3. Lagrangian modeling of small river plumes

An important role of buoyancy force and density gradients are key features of dynamics of small river plumes, which is substantially different from dynamics of ambient sea. Small plumes are characterized by sharp spatial gradients and high temporal variability, while ambient sea has more stable and homogenous structure. Thus, an Eulerian approach denoted by state equations for a fixed point of space is suitable for modeling of complex dependences and feedbacks of sea dynamics, but exhibits certain difficulties if applied for modeling of small plumes. On the other

#### Figure 1.

Schematic of the forces applied to an individual parcel of a river plume.

hand, a Lagrangian approach denoted by state equations for a moving parcel of substance is more efficient for modeling of dynamically active processes and coherent structures typical for small plumes.

We developed a Lagrangian model called Surface-Trapped River Plume Evolution (STRiPE) for simulating the spreading of small river plumes and the associated transport of river-borne suspended sediments [45, 48]. STRiPE represents a river plume as a set of Lagrangian parcels or homogeneous water columns extending from the surface down to the boundary between the plume and the subjacent sea, while their horizontal sizes are presumed to be relatively small. These parcels are initially released from the river mouth and represent river runoff inflowing to sea. The subsequent motion of a parcel is determined by the momentum equation applied to this specific parcel. The overall set of parcels represents the river plume at every step of the model. Thus, the temporal evolution of the plume is simulated. We presume that the buoyant plume remains confined to the surface layer; therefore, the model describes the 2D motion of the parcels, although all parcels exhibit vertical mixing with subjacent sea water. Salinity and density of an individual parcel change in time until it eventually dissipates.

Motion equations, which are applied to an individual parcel, reproduce the main forces that determine river plume dynamics, namely, the Coriolis force, the pressure gradient force, the wind stress force, the friction at the lower boundary of the plume, and the lateral friction (Figure 1). At every step of the computation, the model reads the corresponding values from the input time series of river discharge rate, wind stress, and ambient sea current velocity data. Then, the model calculates the acceleration components (ax, ay) and the resulting velocity components (u, v) for the whole set of parcels. The STRiPE model tracks individual parcels and, therefore, does not use any spatial grid for solving the motion equations. However, an auxiliary horizontal grid with the respective increments Δx and Δy in zonal and meridional directions is used to calculate the spatial derivatives necessary for parameterizing the pressure gradient force and lateral friction applied to the parcel. Continuous fields of velocity, depth, and density within the plume are obtained by extrapolating the respective values from the overall set of parcels.

The momentum equations for an individual parcel are the following:

$$\begin{split} \mathbf{d}^{\mathbf{x}}\_{\mathbf{x}} &= \mathbf{f}\mathbf{f}^{\mathbf{x}}\_{\mathbf{x},\mathbf{y}} + \frac{\mathbf{r}^{\mathbf{i}}\_{\mathbf{x}}}{\mathbf{p}^{\mathbf{i}}\_{\mathbf{x},\mathbf{y}}\mathbf{h}^{\mathbf{i}}\_{\mathbf{x},\mathbf{y}}} - \frac{\mathbf{h}^{\mathbf{i}}\_{\mathbf{x},\mathbf{y}}\mathbf{w}^{\mathbf{i}}\_{\mathbf{x},\mathbf{y}} - \mathbf{u}^{\mathbf{i}}\_{\mathbf{x}\mathbf{a}}}{\mathbf{h}^{\mathbf{i}}\_{\mathbf{x},\mathbf{y}}} \\ &+ \frac{\mathbf{h}\_{\mathbf{h}}}{\mathbf{h}^{\mathbf{i}}\_{\mathbf{x},\mathbf{y}}} \left( \frac{\left(\mathbf{u}^{\mathbf{i}}\_{\mathbf{x}+\mathbf{A},\mathbf{y}} - \mathbf{u}^{\mathbf{i}}\_{\mathbf{x},\mathbf{y}}\right) - \left(\mathbf{u}^{\mathbf{i}}\_{\mathbf{x},\mathbf{y}} - \mathbf{u}^{\mathbf{i}}\_{\mathbf{x}-\mathbf{A},\mathbf{y}}\right)}{\Delta x} + \frac{\left(\mathbf{u}^{\mathbf{i}}\_{\mathbf{x},\mathbf{y}} - \mathbf{u}^{\mathbf{i}}\_{\mathbf{x},\mathbf{y}}\right)}{\Delta y} \right) \\ &- \mathbf{g}^{\mathbf{i}}\_{\mathbf{x}+\mathbf{A},\mathbf{y}}\rho^{\mathbf{i}}\_{\mathbf{x}+\mathbf{A},\mathbf{y}} \left(\rho\_{\mathbf{ez}\alpha} - \rho^{\mathbf{i}}\_{\mathbf{x}+\mathbf{A},\mathbf{y}}\right) - \mathbf{h}^{\mathbf{i}}\_{\mathbf{x}-\mathbf{A},\mathbf{y}}\rho^{\mathbf{i}\_{\mathbf{x},\mathbf{y}}} \left(\rho\_{\mathbf{ez}\alpha} - \rho^{\mathbf{i}}$$

where ux,y; vx,<sup>y</sup> � � are the interpolated velocity components at the x; y � � grid node, f is the Coriolis parameter, τx; τ<sup>y</sup> � � are the wind stress components, ρ is the density of water in the parcel, h is the height of the parcel, ρx,<sup>y</sup> is the interpolated density at the x; y � � grid node, hx,<sup>y</sup> is the interpolated height at the x; y � � grid node, μh, μ<sup>v</sup> are the horizontal and vertical eddy viscosity coefficients, uð Þ sea; vsea are the ambient sea currents velocity components, ρsea is the ambient sea water density, and g is the gravity acceleration. The superscripts denote the model time steps. The first term in Eq. (1) denotes the Coriolis force, the second term stands for the wind stress, the third and fourth terms denote the bottom and lateral friction, and the fifth term stands for the pressure gradient force. After the acceleration components ax; ay � � are obtained from the momentum equations, the final velocities (u, v) for the period (t,t þ Δt) are calculated from kinematic formulas:

$$\begin{aligned} \mathbf{u}^{i+1} &= \mathbf{u}^{i} + \mathbf{a}\_{\mathbf{x}}^{i+1} \Delta t, \\ \mathbf{v}^{i+1} &= \mathbf{v}^{i} + \mathbf{a}\_{\mathbf{x}}^{i+1} \Delta t. \end{aligned} \tag{2}$$

In order to simulate the small-scale horizontal turbulent mixing, the deterministic approach described above was complemented by the random-walk Monte Carlo method [49]:

$$\begin{aligned} \mathbf{x}^{\rm i+1} &= \mathbf{x}^{\rm i} + \mathbf{u}^{\rm i+1} \Delta t - \frac{\mathbf{a}\_{\mathbf{x}}^{\rm i+1} \Delta t^2}{2} + \sqrt{2 \mathbf{D}\_{\mathbf{h}}^{\rm i} \Delta t} \; \eta\_{\mathbf{x}\nu} \\\\ \mathbf{y}^{\rm i+1} &= \mathbf{y}^{\rm i} + \mathbf{v}^{\rm i+1} \Delta t - \frac{\mathbf{a}\_{\mathbf{y}}^{\rm i+1} \Delta t^2}{2} + \sqrt{2 \mathbf{D}\_{\mathbf{h}}^{\rm i} \Delta t} \; \eta\_{\mathbf{y}^{\rm i}} \end{aligned} \tag{3}$$

where x; y � � are the coordinates of an individual parcel, Δt is the time step, Dh is the horizontal diffusion coefficient depending on the velocity field as specified below, and ηx, η<sup>y</sup> are the independent random variables with standard normal

distribution. The horizontal diffusion coefficient used above was calculated from the Smagorinsky formula [50]:

$$\mathbf{D}\_{\mathbf{h}}^{\mathrm{i}} = \mathsf{\zeta}\_{\mathrm{h}} \Delta \mathbf{x} \Delta \mathbf{y} \bigg| \begin{aligned} & \quad \left( \frac{\mathbf{u}\_{\mathrm{x} + \Delta \mathbf{x}, \mathbf{y}}^{\mathrm{i}} - \mathbf{u}\_{\mathrm{x} - \Delta \mathbf{x}, \mathbf{y}}^{\mathrm{i}}}{\Delta \mathbf{x}} \right)^{2} \\ & + \frac{1}{2} \Big( \frac{\mathbf{v}\_{\mathrm{x} + \Delta \mathbf{x}, \mathbf{y}}^{\mathrm{i}} - \mathbf{v}\_{\mathrm{x} - \Delta \mathbf{x}, \mathbf{y}}^{\mathrm{i}}}{\Delta \mathbf{x}} + \frac{\mathbf{u}\_{\mathrm{x}, \mathbf{y} + \Delta \mathbf{y}}^{\mathrm{i}} - \mathbf{u}\_{\mathrm{x}, \mathbf{y} - \Delta \mathbf{y}}^{\mathrm{i}}}{\Delta \mathbf{y}} \right)^{2} + \\ & + \Big( \frac{\mathbf{v}\_{\mathrm{x}, \mathbf{y} + \Delta \mathbf{y}}^{\mathrm{i}} - \mathbf{v}\_{\mathrm{x}, \mathbf{y} - \Delta \mathbf{y}}^{\mathrm{i}}}{\Delta \mathbf{y}} \Big)^{2}, \end{aligned}$$

where ζ<sup>h</sup> is the scaling coefficient.

The simulation of the vertical dissipation of a plume parcel is based on the salinity diffusion equation and assumption that density depends linearly on salinity:

$$\frac{\partial \rho}{\partial t} = \mathbf{D}\_{\mathbf{v}} \frac{\partial^2 \mathbf{S}}{\partial \mathbf{z}^2},\tag{4}$$

or, in a discrete form,

$$
\rho^{\rm i+1} = \rho^{\rm i} + \frac{\mathbf{D}\_{\rm v}^{\rm i}}{\mathbf{h}\_{\rm t}} \frac{\rho\_{\rm sea} - \rho^{\rm i}}{\mathbf{h}^{\rm i}} \Delta t. \tag{5}
$$

where Dv is the vertical diffusion coefficient and ht is the vertical turbulence scale. Hence, as the saline water from the subjacent sea is entrained into the plume gradually replacing the freshwater, the density of water in the parcel increases, while its height decreases according to the following linear equation:

$$\frac{\partial h}{\partial t} = -\frac{\mathbf{D}\_\mathbf{v}^\mathbf{i}}{\mathbf{h}\_\mathbf{t}},\tag{6}$$

or, in a discrete form,

$$\mathbf{h}^{\mathrm{i}+1} = \mathbf{h}^{\mathrm{i}} - \frac{\mathbf{D}\_{\mathrm{v}}^{\mathrm{i}}}{\mathbf{h}\_{\mathrm{t}}} \Delta t. \tag{7}$$

The vertical diffusion coefficient divided by the vertical turbulence scale used above was calculated using the following parameterization based on Richardson number [51]:

$$\frac{\mathbf{D}\_{\mathbf{v}}^{\mathrm{i}}}{\mathbf{h}\_{\mathrm{t}}} = \boldsymbol{\zeta}\_{\mathrm{v}} \boldsymbol{\mu}\_{\mathrm{v}}^{\mathrm{i}} \left(\mathbf{1} - \min\left(\mathbf{1}, \mathrm{Ri}^{\mathrm{i}}\right)^{2}\right)^{3},\tag{8}$$

where <sup>ζ</sup><sup>v</sup> is the scaling coefficient, Ri<sup>i</sup> <sup>¼</sup> Ni<sup>2</sup> Si<sup>2</sup> is the Richardson number,

Ni <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi g ρi ρsea�ρ<sup>i</sup> hi q is the buoyancy frequency, and S<sup>i</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ui �ui ð Þ sea 2 þ vi �vi ð Þ sea 2 q <sup>h</sup><sup>i</sup> is the vertical shear.

Transport and settling of fine suspended sediments discharged from the river mouth is also simulated by STRiPE. In horizontal direction, sediment particle is defined as a passive tracer of a river plume; i.e., the horizontal movement of a

Structure and Dynamics of Plumes Generated by Small Rivers DOI: http://dx.doi.org/10.5772/intechopen.87843

sediment particle is defined by velocity fields calculated within a plume at every modeling step. Vertical movement is calculated individually for every sediment fraction, which have different sizes of particles. For this purpose, we use a combination of a deterministic component defined by sinking of a particle under the gravity force and a stochastic random-walk scheme that reproduces influence of small-scale turbulent mixing. Sediment particles are initially released from the river mouth with river water. During its motion, a particle sinks within the river plume until it reaches the mixing zone between the river plume and the subjacent sea water. After the particle descends beneath the lower boundary of the plume, it is regarded as settled to ambient sea and is stopped to be simulated by STRiPE. The STRiPE is intended to simulate transport of relatively small particles with diameter less than 10�<sup>4</sup> m; therefore, gravity-induced vertical motion is determined by Stokes' law, and particle settling velocity ws is calculated as follows: ws <sup>¼</sup> gd<sup>2</sup> <sup>ρ</sup>s�ρ<sup>i</sup> ð Þ <sup>18</sup>μρ<sup>i</sup> ,

where d is the sediment particle diameter, ρ<sup>s</sup> is the sediment particle density, and μ is the dynamic water viscosity.

The total vertical displacement of a sediment particle determined by gravitational sinking, vertical advection, and turbulent mixing was parameterized by the random-walk Monte Carlo method, which represents features of spatially nonuniform turbulent mixing:

$$
\Delta \mathbf{z} = \left( \mathbf{w}\_s + \frac{\partial \mathbf{K}}{\partial \mathbf{z}} \right) \Delta t + \sqrt{\frac{2}{3} \mathbf{K}\_\mathbf{v} \left( \mathbf{z} + \frac{1}{2} \frac{\partial \mathbf{K}\_\mathbf{v}}{\partial \mathbf{z}} \Delta t \right)} \Delta t \eta\_\mathbf{v} \tag{9}
$$

where Δz is the vertical displacement of a particle, Kv is the vertical diffusion coefficient, and η is a random process with standard normal distribution.

The main advantage of STRiPE lies in its computational efficiency in simulating spreading and mixing of river plumes as compared to Eulerian models. However, STRiPE does not reproduce any influence of a river plume on the ambient ocean, which is an important issue for large river plumes. Thus, STRiPE should be applied for simulation of spreading and mixing of small river plumes that limitedly influence the ambient sea.
