Towards a parametrization of river discharges into ocean general circulation models: a closure through energy conservation

Abstract

Diagnostic methods are defined in order to compare two numerical simulations of ocean dynamics in a region of freshwater influence. The first one is a river plume simulation based on a high resolution numerical configuration of the POM coastal ocean model in which mixing parametrizations have been previously defined. The second one is a simulation based on the NEMO Global Ocean Model used for climate simulations in its half-a-degree configuration in which a river inflow is represented as precipitation on two coastal grid cells. Both simulations are forced with the same freshwater inflows and wind stresses. The divergence of volumetric fluxes above and below the halocline are compared. Results show that when an upwelling wind blows, the two models display similar behavior although the impact of lack of precision can be observed in the NEMO configuration. When a downwelling wind blows, the NEMO Global Ocean configuration can not reproduce the coastally trapped baroclinic dynamics because its grid resolution is too coarse. To find a parametrization to help represent these dynamics in ocean general circulation models, a method based on energy conservation is investigated. This method shows that it is possible to link the energy fluxes provided by river inflows to the divergence of energy fluxes integrated over the grid cells of ocean general circulation models. A parametrization of the dynamics created by freshwater inflows is deduced from this method. This enabled creation of a box model that proved to have the same behavior as the fluxes previously computed from the high resolution configuration.

Appendices

Appendix A: Notations

Tables 1, and 2.

Table 1 Variables and parameters of the box model used to estimate the coastal overturning created by a river inflow

Table 2 Definition of the parameters to set the box model used to estimate the coastal overturning created by the Mekong river

Appendix B: Mode splitting

The grid cell of the OGCM in which a river inflow is injected is a region of freshwater influence. However, a large scale current may flow in this region and could flow even without any freshwater input in the grid cell. This large scale current may be caused by large scale baroclinic effects or wind forcings, but we know from Hetland (2005) that it is very unlikely that the river plume halocline can interact with the thermocline. So whatever the type of circulation in the coastal area, the large scale currents can be considered as being barotropic. The river plume is considered as a single baroclinic mode.

The pressure perturbation can be split into two parts: one related to the barotropic mode and the other to the baroclinic mode. The same is true of the velocity field

$$ p^{\prime}=p_{\rm trop}^{\prime}+p_{\rm clin}^{\prime} \quad {\user2{u}} = {\user2{u}}_{{\rm trop}}+{\user2{u}}_{{\rm clin}}. $$

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We also define the x _l (longshore) axis unit vector. As the river plume is coastally trapped, we suppose that u _clin = u _clin x _l . Considering that the barotropic component is also mostly alongshore in the area where it may interact with the plume yields: u _trop = u _trop x _l .

Integrating the divergence of F _tot over the control volume yields Eq. 21

$$ \begin{aligned} \int\int\limits_{S_{\rm bound}} {\user2{F}}_{\rm tot} \cdot {\user2{dS}}_{{\rm bound}} &= {\left. \begin{array}{l}\int\int\limits_{S_{\rm bound}} \left[\frac{1}{2} \rho (u_{\rm clin}^3 + 3 u_{\rm clin}^2 u_{\rm trop}) \right.\\ + \left.(p_{\rm clin}^{\prime} + \rho^{\prime} \Phi) (u_{\rm clin}+u_{\rm trop})\right] {\user2{x}}_{\user2{l}} \cdot {\user2{dS}}_{\rm bound} \end{array}\right\}} P_{\rm clin} \\ &\quad + {\left.\begin{array}{l}\int\int\limits_{S_{\rm bound}} \left[\frac{1}{2} \rho (3 u_{\rm trop}^2 u_{\rm clin} + u_{\rm trop}^3)\right.\\ + \left. p_{\rm trop}^{\prime} (u_{\rm trop} + u_{\rm clin})\right] {\user2{x}}_{\user2{l}} \cdot {\user2{dS}}_{\rm bound}\end{array}\right\}} P_{\rm trop}. \end{aligned} $$

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P _clin and P _trop are respectively the divergence of energy fluxes of the baroclinic and barotropic modes integrated over the control volume. For each power term, we can recognize the kinetic energy of a given mode as being advected by the mode itself or by the adjacent mode. In the case of the Mekong river plume, the ambient shelf circulation is wind driven only. This means that during a downwelling event, the baroclinic velocities will be of a higher magnitude than the barotropic ones, lowering the non-linear interactions. However, from a more general perspective, if the ambient shelf circulation is of an equivalent magnitude, non-linear interactions may appear.

Appendix C: Power contributions

1.1 Appendix C.1: Power provided by the river

The main power source that exists even in the absence of downwelling wind is the power provided by the river. It is the power supply of the pump described in Fig. 6. The power provided by the river is expressed using Eq. 5. Assuming a rectangular section of the estuarine mouth yields an expression that is simple to evaluate for the power P _r provided by the river:

$$ P_r= (\rho_o-\rho_r) g h_{\rm est} Q_r + \frac{1}{2} \rho u^3 h_{\rm est} B. $$

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In Eq. 22, ρ _r is the freshwater density and we consider the width B and the depth h _est of all the estuaries as constant. In this particular case of a simple output of freshwater, the expressions for the fluxes of geopotential and the power of pressure forces are the same. In most cases, the kinetic part of the power is very small compared with the flux of geopotential and the work of pressure forces (Garvine 1999) and will dissipate anyway. Therefore Eq. 22, which works for a single estuary can also work for a river delta: assuming that all river mouths have the same depth, P _r has the same expression whatever the number of river mouths where Q _r is the total freshwater discharge to the coastal area.

1.2 Appendix C.2: Power through side boundaries

We define the following powers representing respectively the divergence of energy fluxes above and below the halocline:

$$ P_a= \int\int\limits_{S_{\rm bound/above}} {\user2{F}}_{\rm tot} \cdot {\user2{dS}}_{\rm bound} \quad P_b= \int\int\limits_{S_{\rm bound/below}} {\user2{F}}_{\rm tot} \cdot {\user2{dS}}_{\rm bound}. $$

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Appendix C.3: Surface boundary: wind power

Downwelling wind has an influence on the river plume (Chao 1988; Fong et al. 1997; Fong and Geyer 2002; Berdeal-Garcia et al. 2002; Whitney and Garvine 2005). However, wind provides power to all water columns including those which are not in the river plume. In the case of a downwelling wind, a storm surge coastal jet may be created (Gill 1982) that exists even in the absence of a river plume. This coastal jet is an ambient current, and is scaled by a barotropic Rossby radius. This scale is much bigger than its baroclinic equivalent, and is therefore represented by an OGCM in most cases.

Wind blows over the entire surface of the control domain. However, wind power transmitted at the surface of the river plume is captured by the motion of the baroclinic mode only, and wind power transmitted at the surface of the area outside the river plume is captured by the motion of the barotropic mode only (see Appendix B). We define P _w−use as the total wind power transmitted to the motion in the control volume. P _w−use = P _w−use/trop + P _w−use/clin where P _w−use/trop and P _w−use/clin are, respectively, the wind powers transmitted to barotropic and baroclinic motions.

If we suppose a quasi-permanent state, the barotropic Kelvin wave is in equilibrium with its wind forcing and bottom friction power P _F (Gill 1982):

$$ P_{w-{\rm use/trop}}+ P_{F}=0. $$

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P _F is a difficult term to estimate because we do not consider any heat balance for the control volume, but assuming that wind power balances friction means that there is a balance between all powers for the barotropic mode. The Bernouilli equation can thus be applied. For a permanent state, the power balance for the barotropic mode is reached. Hence its divergence of energy fluxes P _trop is zero.

For this reason, we do not take into account the wind power transmitted at the surface of the control volume outside the river plume. This is why the computation of P _a and P _b defined previously can be done by integrating F _tot over the vertical side boundaries of the control volume, whether they are influenced by the river plume or not.

In order to compute P _w−use/clin, we first compute the total power transmitted to the river plume motion P _w . The wind power transmitted to the river plume can be expressed as follows in Eq. 25 in which S _Plume–Surf stands for the surface of the river plume, u is the sea surface velocity and ${\varvec{\tau}}$ is the surface wind stress.

$$ P_w= \int\int\limits_{S_{{\rm Plume}-{\rm Surf}}} {\varvec{\tau}} \cdot {\user2{u}} \; dS_{\rm Surf}. $$

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Following the work of Kundu (1980) and Klein and Coantic (1981), we consider that the dissipation of the wind power E along the water column over the halocline can be expressed as:

$$ E = {\varvec{\tau}} . ({\user2{u}} - {\overline{\user2{u}}}). $$

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In Eq. 26, ${\overline{\user2{u}}}$ is the mean velocity above the halocline on a given water column. This yields an expression for P _w−use/clin:

$$ P_{w-{\rm use/clin}}= \int\int\limits_{S_{{\rm Plume}-{\rm Surf}}} {\varvec{\tau}} \cdot {\overline{\user2{u}}} \; dS_{\rm Surf}. $$

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2.1 Appendix C.4: Power lost by geostrophic adjustment

Gill (1982), Sect. 7.2 considers the case of a barotropic adjustment from an initial discontinuity in sea surface height. He shows that in the case of a rotational frame, a barotropic Kelvin wave establishes but that its kinetic energy is only 1/3 of the difference in potential energy between the initial state and the establishment of a barotropic Kelvin wave at time ${\gg}1/f .$ Our case is very similar except that we do not consider a barotropic case but a baroclinic one, and that our initial condition is not still, but is a flux of baroclinic discontinuity unaffected by rotational effects. Therefore, gravity waves have to be generated by the geostrophic adjustment, and propagate away from the region of the estuarine mouth. It is very unlikely that we are able to capture the energy carried out by these gravity waves on the vertical side boundaries of the control volume because our daily outputs of the river plume simulation do not have a frequency high enough (nor do OGCMs used in climate modeling). Moreover, we do not want to mix this energy flux with the energy fluxes generated by the phenomenon that is most important for us (i.e.: the river plume). Meanwhile, we have to take into account the fact that a power $P_{I_{\rm wav}}$ leaves the control volume without participating in the coastal overturning. We consider that $P_{I_{\rm wav}}$ takes out of the domain 2/3 of the kinetic energy that has been created by the geostrophic adjustment of the low-density input, which means twice the kinetic energy that leaves the domain with the surface current: this means the kinetic part of P _clin. The value of 2/3 can be chosen as the halocline of a river plume is usually very sharp so the plume may be considered as a two layer front system (Ou 1986).

This yields Eq. 28:

$$ P_{I_{\rm wav}}= 2 \int\int\limits_{S_{\rm bound/above}} \frac{1}{2} \rho (u_{\rm clin}^3 + 3 u_{\rm clin}^2 u_{\rm trop}) {\user2{x}}_{\user2{l}} \cdot {\user2{dS}}_{\rm bound}. $$

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Of course, this is true only if the scale of the control volume is large compared with the scale of the Kelvin wave. This happens to be the case as what OGCMs lack to resolve river plumes properly is that their grid cell resolution is too coarse compared with the baroclinic Rossby radius of the density anomaly.

2.2 Appendix C.5: Energy loss through production of heat

P_diab represents the loss of energy by diabatic processes generated by the motion of the river plume. It is a difficult term to estimate. But we know from river plume dynamics that the kinetic energy contributed by the freshwater inflow, and the kinetic energy contributed by the undercurrent both vanish. As a first approximation, we can set P_diab to the opposite of the kinetic energy contribution of the river and of the undercurrent. However, some extra terms are required if a box model is to be created.