Tracer equations

The general conservation equation for tracers $c^i$ with $1\leq i \leq N_c$ (with $N_c$ being the number of tracers), which can e.g. be temperature, salinity, nutrients, phytoplankton, zoo-plankton, suspended matter, chemical concentrations etc. is given as:

$$\begin{array}{l} \partial_t c^i +\partial_x (uc^i) +\partial_... ...tial_x c^i) -\partial_y(A_h^T \partial_y c^i) =Q^i. \end{array}$$ (97)

Here, $\nu'_t$ denotes the vertical eddy diffusivity and $A_h^T$ the horizontal diffusivity. Vertical migration of concentration with migration velocity $w_s^i$ (positive for upward motion) is considered as well. This could be i.e. settling of suspended matter or active migration of phytoplankton. In order to avoid stability problems with vertical advection when intertidal flats are drying, the settling of SPM is linearly reduced towards zero when the water depth is between the critical and the minimum water depth. This is done by means of multiplication of the settling velocity with $\alpha$, (see the definition in equation (5)). $Q^i$ denotes all internal sources and sinks of the tracer $c^i$. This might e.g. be for the temperature equation the heating of water due to absorption of solar radiation in the water column.

Surface of bottom boundary conditions for tracers are usually given by prescribed fluxes:

$$-\alpha w_s^i c^i+\nu_t' \partial_z c^i = F^i_s z=\zeta$$ (98)

and

$$-\alpha w_s^i c^i+\nu_t' \partial_z c^i = -F^i_b z=-H,$$ (99)

with surface and bottom fluxes $F^n_s$ and $F^n_b$ directed into the domain, respectively.

At open lateral boundaries, the tracers $c^n$ are prescribed for the horizontal velocity normal to the open boundary flowing into the domain. In case of outflow, a zero-gradient condition is used.

All tracer equations except those for temperature, salinity and suspended matter will be treated in the future by means of GOTM-BIO.

The two most important tracer equations which are hard-coded in GETM are the transport equations for potential temperature $\theta$ in $^{\circ}$C and salinity $S$ in psu (practical salinity units):

$$\begin{array}{l} \partial_t \theta +\partial_x (u\theta) +\pa... ...artial_y \theta) =\frac{\partial_z I}{c'_p \rho_0}, \end{array}$$ (100)

$$\begin{array}{l} \partial_t S +\partial_x (uS) +\partial_y(vS... ...S \partial_x S) -\partial_y(A_h^S \partial_y S) =0. \end{array}$$ (101)

On the right hand side of the temperature equation (100) is a source term for absorption of solar radiation with the solar radiation at depth $z$, $I$, and the specific heat capacity of water, $c'_p$. According to Paulson and Simpson (1977) the radiation $I$ in the upper water column may be parameterised by

$$I(z) = I_0 \left(ae^{-\eta_1z}+(1-a)e^{-\eta_2z}\right).$$ (102)

Here, $I_0$ is the albedo corrected radiation normal to the sea surface. The weighting parameter $a$ and the attenuation lengths for the longer and the shorter fraction of the short-wave radiation, $\eta_1$ and $\eta_2$, respectively, depend on the turbidity of the water. Jerlov (1968) defined 6 different classes of water from which Paulson and Simpson (1977) calculated weighting parameter $a$ and attenuation coefficients $\eta_1$ and $\eta_2$.

At the surface, flux boundary conditions for $T$ and $S$ have to be prescribed. For the potential temperature, it is of the following form:

$$\nu'_t \partial_z T= \frac{Q_s+Q_l+Q_b}{c'_p \rho_0}, z=\zeta,$$ (103)

with the sensible heat flux, $Q_s$, the latent heat flux, $Q_l$ and the long wave back radiation, $Q_b$. Here, the Kondo (1975) bulk formulae have been used for calculating the momentum and temperature surface fluxes due to air-sea interactions. In the presence of sea ice, these air-sea fluxes have to be considerably changed, see e.g. Kantha and Clayson (2000b). Since there is no sea-ice model coupled to GETM presently, the surface heat flux is limited to positive values, when the sea surface temperature $T_s$ reaches the freezing point

$$T_f=-0.0575\,S_s+1.710523\cdot 10^{-3}\, S_s^{1.5} -2.154996\cdot 10^{-4}\,S_s^2\approx -0.0575\,S_s$$ (104)

with the sea surface salinity $S_s$, see e.g. Kantha and Clayson (2000a):

$$Q_{surf} = \left\{ \begin{array}{ll} Q_s+Q_l+Q_b, & \mbox{ fo... ...\ \\ \max\{0,Q_s+Q_l+Q_b\}, & \mbox{ else.} \end{array}\right.$$ (105)

For the surface freshwater flux, which defines the salinity flux, the difference between evaporation $Q_E$ (from bulk formulae) and precipitation $Q_P$ (from observations or atmospheric models) is calculated:

$$\nu'_t\partial_z S = \frac{S(Q_E-Q_P)}{\rho_0(0)}, z=\zeta,$$ (106)

where $\rho_0(0)$ is the density of freshwater at sea surface temperature. In the presence of sea-ice, the calculation of freshwater flux is more complex, see e.g. Large et al. (1994). However, for many short term calculations, the freshwater flux can often be neglected compared to the surface heat flux.

A complete revision of the surface flux calculation is currently under development. It will be the idea to have the same surface flux calculations for GOTM and GETM. In addition to the older bulk formulae by Kondo (1975) we will also implement the more recent formlations by Fairall et al. (1996).

Heat and salinity fluxes at the bottom are set to zero.