8.5 River runoff

River runoff may be an important source of fresh water. On a glance the most realistic and rigorous implementation could be an open boundary condition with prescribed values for the mass transport, the heat flux and the other tracer fluxes. However, such an implementation of rivers is overly complex and not necessary for the most purposes. Recall that for the momentum and tracer time tendency of a surface grid box, it is unimportant whether a flux enters the box vertically through the sea surface or horizontally through a vertical cell face. Thus, the fresh water flux into the surface boxes in a river mouth can be supplemented with the fresh water flux of the river. Similarly the corresponding tracer and momentum fluxes can be added to the surface boundary fluxes.

Data on river discharge are provided as a volume flux R(t) as function of time given, e.g., in units $\mbox{m}^3\,\mbox{s}^{-1}$. This flux has to be distributed over the area A_R of one or more surface boxes, where the river flux enters the model. In this surface boxes the river flux is,

$$\frac{R(t)}{A_R}.$$ q_R = (8.25)

Obviously, the surface integral over q_R gives the total river volume flux rate R(t). q_R has the dimension of a velocity and has to be added to the surface fresh water flux velocity,

$$\rightarrow$$ q_w q_w + q_R. (8.26)

Along with the fresh water, heat and additional tracers are entrained. The corresponding fluxes

$$\frac{R(t)}{A_R} T_R$$ Q_RT =

$$q_R \, T_R,$$ = (8.27)

where T_R is the tracer concentration in the river water, must be added to the fresh water driven surface fluxes,

$$Q_{wT} \rightarrow Q_{wT} + Q_{RT},$$ (8.28)

The additional momentum entrained with rivers can be estimated from the river volume flux rate, R(t), and the vertical river cross section, C_R. When the cross river circulation in the boundary cells is zero the average velocity is through the vertical cross section is

$${\bf u}_R \frac{R(t)}{\rho C_R}\, \tilde{\bf C}_R.$$ = (8.29)

$\tilde{\bf C}_R$ is the normal unit vector of the river cross section, C_R. The momentum advected through the river cross section with the river flux is

$${\bf M}_R \rho \, C_R \, \vert{\bf u}_R\vert\, {\bf u}_R$$ =

$$\rho\, R(t) \, {\bf u}_R.$$ = (8.30)

The components of the equivalent surface stress, which gives the same momentum input in form of a vertical momentum flux, are

$$\tau^{\lambda}_R \rho\, \frac{R(t)}{A_R}\,u_R ,$$ =

$$\rho\,q_R\, u_R,$$ = (8.31)

$$\tau^{\phi}_R \rho\,q_R\, v_R.$$ = (8.32)

As such, rivers can be included without modifications of the basic code. The only information required is the river discharge R(t), the tracer concentration in the river T_R and the river cross section $C_R\, \tilde{\bf C}_R$. The fluxes q_R, Q_RT and $\tau_R$ are added simply to the other fresh water induced surface fluxes.

For many purposes this set of information is not available completely and approximations are necessary. For the river salinity the assumption s_R = 0should be well justified. To find values for an unknown river temperature $\theta_R$, it can be a guideline, that the approximation $\theta_R\approx \theta_1$ leaves the sea surface temperature in the river cells unchanged. To illustrate this consider a grid cell at rest changed only by fresh water flux due to a river. In this case the sea surface increases as

$$\eta_t$$ = q_R. (8.33)

The remainder of the tracer equation is

$$\partial_{t} \, T_1 \frac{q_R T_R - T_{1} \, \eta_{t}}{\eta -z_{1}},$$ =

$$\frac{q_R \,(T_R - T_{1})}{\eta -z_{1}}.$$ = (8.34)

For salinity, with T_R = s_R = 0, there is a negative time tendency in the surface salinity due to the river inflow. For other tracers, the time tendency vanishes for T_R = T₁. Hence, in the general case the approximation T_R = T₁ seems to be the most natural assumption to fill gaps in the river data base.

The same arguments apply for the momentum. If the river cross section is unknown, the assumption ${\bf u}_R = {\bf u}_1$ could be reasonable, i.e., the river discharge has the same average momentum as a model surface cell. In this case, the river increases the sea surface height as well as the total momentum of surface cells, but the average velocity of a surface cells is not changed. This holds also for the linearized free surface approximation.

The treatment of river inflow with the aforementioned method has some limitations and needs some remarks.

• Data on river runoff are provided mostly as climatological data, i.e. only monthly or sesonal time scales are resolved. However, big rivers may have estuaries with a large storage capacity. Due to wind surge or tides, the river outflow may be temporarily blocked and fresh water enters the ocean intermittently in the form of drifting river plumes. To implement such features, the processes producing intermittency in the river outflow must be resolved by the model. Thus, the estuary must be part of the model and the river runoff must enter the model area more upstream.
• The river is assumed to be unstratified and horizontally uniform. Improvements require a higher river model resolution.
• River inflow is confined to the surface cells. For rivers much deeper than the surface layer, this may introduce too much stratification and an intensified estuarine circulation. If this is not desired, two or more surface layers should be mixed explicitly at the cells with river inflow. This requires code changes and is not part of the basic MOM code.
• The momentum flux may be incorrect. The method distributes the mass flux over several tracer points and establishes surface pressure gradients which reflect the model setup but not the dynamics in a river mouth. If the dynamical features of the outflow are important, a sufficiently large part of the river must be included in the model. In order to prevent unrealistic currents due to different position of velocity and tracer cells in the Arakawa B-grid, the fresh water flux and the tracer flux should be distributed at least over two adjacent tracer cells, the momentum flux should be added in the corresponding velocity cell between the tracer cells. Figures 8.1 and 8.2 give some hints how the rivers could be configurated. However, the method does not require an adjustment of the coastlines to configure rivers as long as landpoints or nonadvective cells are avoided.