**2. The numerical model**

The numerical model is based on motion and continuity equations (Liu & Leendertse, 1978) tested by the authors. The model utilizes a Liu and Leendertse's scheme describing vertical water motion by calculation of the turbulence field. The model equations for conservation of momentum and continuity, written in Cartesian coordinates, in incompressible fluid conditions and under the effects of Earth's rotation are:

$$\frac{\partial \mathbf{u}}{\partial t} + \frac{\partial (\mu \mathbf{u})}{\partial \mathbf{x}} + \frac{\partial (\mu \mathbf{v})}{\partial y} + \frac{\partial (\mu \mathbf{w})}{\partial z} - f \cdot \boldsymbol{\nu} + \frac{1}{\rho} \frac{\partial p}{\partial \mathbf{x}} - \frac{1}{\rho} \left( \frac{\partial \sigma\_{\mathbf{x}}}{\partial \mathbf{x}} + \frac{\partial \tau\_{\mathbf{xy}}}{\partial y} + \frac{\partial \tau\_{\mathbf{xz}}}{\partial z} \right) = \mathbf{0} \tag{1}$$

$$
\sigma\_{\rm x} = A\_{\rm x} \frac{\partial \mathbf{u}}{\partial \mathbf{x}}; \qquad \tau\_{\rm xy} = A\_{\rm x} \frac{\partial \mathbf{u}}{\partial \mathbf{y}} \tag{2}
$$

$$\frac{\partial \mathbf{v}}{\partial t} + \frac{\partial (vu)}{\partial \mathbf{x}} + \frac{\partial (vv)}{\partial y} + \frac{\partial (vw)}{\partial z} + f \cdot u + \frac{1}{\rho} \frac{\partial p}{\partial y} - \frac{1}{\rho} \left( \frac{\partial \tau\_{yx}}{\partial \mathbf{x}} + \frac{\partial \sigma\_y}{\partial y} + \frac{\partial \tau\_{yz}}{\partial z} \right) = 0 \tag{3}$$

$$
\sigma\_y = A\_y \frac{\partial \upsilon}{\partial y}; \qquad \tau\_{yx} = A\_y \frac{\partial \upsilon}{\partial y} \tag{4}
$$

$$\frac{\partial w}{\partial t} + \frac{\partial (wu)}{\partial x} + \frac{\partial (wv)}{\partial y} + \frac{\partial (ww)}{\partial z} + \frac{1}{\rho} \frac{\partial p}{\partial z} - \frac{1}{\rho} \left( \frac{\partial \tau\_{zx}}{\partial x} + \frac{\partial \tau\_{zy}}{\partial y} + \frac{\partial \sigma\_z}{\partial z} \right) + g = 0 \tag{5}$$

$$\frac{\partial u}{\partial x} + \frac{\partial v}{dy} + \frac{\partial w}{dz} = \mathbf{0} \tag{6}$$

where *t* denotes time, x, y and *z* are Cartesian coordinates (positive towards Est, South, up) *u, v, w* denote velocity components in the direction of *x,* y, z, *f* is the Coriolis parameter (assumed to be a constant), *g* is the acceleration due to gravity, ρ is the density of water, *A* is the horizontal eddy viscosity σi, τij with i, j = x, y, z components of Reynolds tensor proportional to vertical gradient of velocity. In accordance with other authors it was assumed as adequate the use of two-dimensional depth-integrated equations for

time. The investigated area relates to the coastal zone near Cesenatico (Adriatic Sea, Italy). The aim of this chapter is to describe the dynamic of freshwater dispersion and to show the results of the simulation of flushing, mixing and dispersion of discharged freshwater from


The numerical model is based on motion and continuity equations (Liu & Leendertse, 1978) tested by the authors. The model utilizes a Liu and Leendertse's scheme describing vertical water motion by calculation of the turbulence field. The model equations for conservation of momentum and continuity, written in Cartesian coordinates, in incompressible fluid

1 1 <sup>0</sup> *xy <sup>x</sup> xz u uu uv uw <sup>p</sup> f v tx y z x xyz*

; *xy x*

1 1 <sup>0</sup> *v vu vv vw <sup>p</sup> yx y yz f u tx y z y xyz*

; *yx y*

1 1 <sup>0</sup> *zy zx <sup>z</sup> w wu wv ww <sup>p</sup> <sup>g</sup> t x y z z xyz*

<sup>0</sup> *uvw x dy dz* 

where *t* denotes time, x, y and *z* are Cartesian coordinates (positive towards Est, South, up) *u, v, w* denote velocity components in the direction of *x,* y, z, *f* is the Coriolis parameter (assumed to be a constant), *g* is the acceleration due to gravity, ρ is the density of water, *A* is the horizontal eddy viscosity σi, τij with i, j = x, y, z components of Reynolds tensor proportional to vertical gradient of velocity. In accordance with other authors it was assumed as adequate the use of two-dimensional depth-integrated equations for

 

 

 

 

 

 

*<sup>v</sup> <sup>A</sup> y* 

*<sup>u</sup> <sup>A</sup> y* 

 

 

(2)

(4)

 

(6)

(1)

(3)

(5)

the harbour channel mouth under different forcing conditions. The chapter will be organized in the following sections:


conditions and under the effects of Earth's rotation are:

() () ( )

() () ( )

*x x <sup>u</sup> <sup>A</sup> x* 

*y y <sup>v</sup> <sup>A</sup> y* 

() () ( )




**2. The numerical model** 

conservation of mass and momentum for a typically well-mixed water column due to wind and tidal stirring. So the vertical momentum equation has been substituted by baroclinic pressure equation (8) where sea water density has been formulated according with the international thermodynamic equation of sea water based on the empirical state function of UNESCO81 which links density to Salinity, Temperature and Pressure:

$$
\frac{\partial S}{\partial t} + \frac{\partial (Su)}{\partial \mathbf{x}} + \frac{\partial (Sv)}{\partial y} + \frac{\partial (Sw)}{\partial z} - \frac{\partial}{\partial \mathbf{x}} \left[ D\_x \frac{\partial S}{\partial \mathbf{x}} \right] - \frac{\partial}{\partial y} \left[ D\_y \frac{\partial S}{\partial y} \right] - \frac{\partial}{\partial z} \left[ k \frac{\partial S}{\partial z} \right] = 0 \tag{7}
$$

$$\frac{\partial p}{\partial z} + \rho(\mathbf{S}, T)\mathbf{g} = \mathbf{0} \tag{8}$$

$$\frac{\partial T}{\partial t} + \frac{\partial \langle Tu \rangle}{\partial \mathbf{x}} + \frac{\partial \langle Tv \rangle}{\partial y} + \frac{\partial \langle Tw \rangle}{\partial z} - \frac{\partial}{\partial \mathbf{x}} \left[ D\_x \frac{\partial T}{\partial \mathbf{x}} \right] - \frac{\partial}{\partial y} \left[ D\_y \frac{\partial T}{\partial y} \right] - \frac{\partial}{\partial z} \left[ k^1 \frac{\partial T}{\partial z} \right] = 0 \tag{9}$$

*S* and T are salinity and temperature, respectively. *Dx ,DY* are horizontal eddy diffusivities for *S* and T; k and k' are vertical diffusion coefficients for mass and wheat. For vertical balance an E coefficient of vertical exchange is introduced for momentum which relates vertical Reynolds forces to the vertical gradient of horizontal components of velocities and expressed by Kolmogorov e Prandtl as:

$$E = \rho L \sqrt{e} \exp^{\left(-m \text{Ri}\right)}\tag{10}$$

$$Ri = -\frac{g}{\rho e} \frac{\partial \mathcal{p}}{\partial z} L^2 \tag{11}$$

$$L = \mathcal{K}z\sqrt{1 - \frac{z}{d}}\tag{12}$$

$$
\sigma\_{xx} = E\_x \frac{\partial u}{\partial z} \tag{13}
$$

$$
\tau\_{yz} = E\_y \frac{\partial v}{\partial z} \tag{14}
$$

where m is a numeric parameter and *Ri* is the Richardson number in terms of vertical gradient of density (ρ) and of eddy kinetic energy (e) and L defined, according with Von Kàrman, by (numerical coefficient) and d distance of bottom (z=0) from surface (z=d). Vertical exchange coefficients for mass (k) and heat (k' ) are defined similarly to E coefficients using adequate parameters substitutive of ρ and m. The adopted vertical scheme introduces eddy kinetic energy as a state function which requires its own dynamic equation for balance and conservation:

$$\frac{\partial \varepsilon}{\partial t} + \frac{\partial (\varepsilon u)}{\partial \mathbf{x}} + \frac{\partial (\varepsilon v)}{\partial y} + \frac{\partial (\varepsilon w)}{\partial z} - \frac{\partial}{\partial \mathbf{x}} \left[ D\_x \frac{\partial \varepsilon}{\partial \mathbf{x}} \right] - \frac{\partial}{\partial y} \left[ D\_y \frac{\partial \varepsilon}{\partial y} \right] - \frac{\partial}{\partial z} \left[ E\_\varepsilon \frac{\partial \varepsilon}{\partial z} \right] + S - D\_\varepsilon = 0 \tag{15}$$

Freshwater Dispersion Plume in the Sea: Dynamic Description and Case Study 133

thermoaline and quality parameters appear strongly conditioned by external waves and currents also in dry weather conditions. Recent monitoring investigations (Mancini, 2009) reveal vertical parameter profiles at mouth varying from an initial uniform vertical profile, also in correspondence with low tidal outgoing ranges. During these conditions, reinforced afternoon wind generates significant sea waves, which oppose the surface freshwater flow. On the contrary, during nightly major outgoing tidal phases, strong stratification is maintained in the mouth zone, as in the dispersion plume area facing the breakwaters. Morphologic, hydraulic and water quality measurements have been executed into the transition estuary of the harbour canal and near the mouth of adjacent breakwaters (a view

A

B Fig. 1. A) Aerial view of Cesenatico. The bullets represent the position of the sample in Fig. 1B: from the city centre to the sea, respectively: Mariner Museum, Garibaldi Bridge, Vincian Ports and sea mouth. B) Salinity and oxygen profiles inside the harbour canal in the position

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 **DEPTH (cm)**

marinery museum

marinery museum

3,0

4,0

5,0

6,0

7,0

**OXYGEN (m g/l)**

8,0

sea mouth

vincian ports

Garibaldi bridge

vincian ports

9,0

of the breakwater is shown in Fig. 2B).

plotted in nearby Fig. 1A.

20,0

**S A LIN ITY (g/l)**

25,0

30,0

Garibaldi bridge

sea mouth

35,0

40,0

45,0

50,0

with horizontal and vertical exchange eddy diffusivities Dx Dy and Ee were defined in a similar way to exchange mass coefficients. So energy sourcing into the grid is detected in the function of strain tensions induced by vertical velocity and the term of energy dissipated by shear stress at the bottom is calculated from energy in flux direction in lower level S and Chézy shear coefficient C. In the surface layer a wind effect generating turbulence by waves is considered in the higher part of the water column. Supposing constant motion for the sea, *Et* is the total energy generated for surface units in the function of wind velocity

$$S = L\sqrt{e} \left(\frac{\partial \overline{u}}{\partial z}\right)^2\tag{16}$$

$$D\_e = a\_2 \frac{e^2}{L} \tag{17}$$

$$S = g \frac{\mathbf{U}^3}{\mathbf{C}^2} \tag{18}$$

$$E\_t = 5.610^{-9} u\_w^4 \tag{19}$$

where *u* is the mean velocity, De the dissipation coefficient, a2 a constant parameter taking account of energy transfer from high to small turbulence conditions and *uw* the wind velocity.

The model equations are discretized and solved into a finite-difference formulation.
