Mathematical formulation

The general structure of a biogeochemical model with $ I$ state variables expressed as ensemble averaged concentrations is given by the following set of equations:

$\displaystyle \partial_t c_i + \partial_z \left(m_i c_i - K_V \partial_z c_i\right) = P_i(\vec{c}) -D_i(\vec{c}), \;\; i = 1,\ldots,I, \quad j,k = 1,\ldots,3,$ (263)

with $ c_i$ denoting the concentrations of state variables. Furthermore, $ m_i$ represents the autonomous motion of the ecosystem component $ c_i$ (e.g. sinking or active swimming), and $ K_V$ represents the eddy diffusivity. The source and sink terms of the ecosystem component $ c_i$ are summarised in $ P_i(\vec{c})$ and $ D_i(\vec{c})$, respectively. For three-dimensional models, advection with the flow field and horizontal advection would have to be accounted for additionally. In many biogeochemical models, some of the state variables have positive lower limits. In order to account for this, we defined all state variables as the difference between the actual value and their lower limit, such that (for non-negative state variables only) the model value $ c_i$ represents a concentration of $ c_i+c_i^{\min}$ where $ c_i^{\min}$ is the lower limit of $ c_i$.

The gradient term on the left hand side of (263) is the total transport, for which typically surface and bottom boundary conditions

$\displaystyle K_V\partial_z c_i\big\vert _{z=\eta} = F^s_i,\qquad K_V\partial_z c_i\big\vert _{z=-H} = -F^b_i,$ (264)

with surface and bottom fluxes, $ F^s_i$ and $ F^b_i$, respectively, are applied. The right hand side denotes the reaction terms, which are composed of contributions $ d_{i,j}(\vec{c})$, which represent reactive fluxes from $ c_i$ to $ c_j$, and in turn, $ p_{i,j}(\vec{c})$ are reactive fluxes from $ c_j$ received by $ c_i$:

$\displaystyle P_i(\vec{c}) = \sum^I_{j=1} p_{i,j}(\vec{c}), \;\;\;\; D_i(\vec{c}) = \sum^I_{j=1} d_{i,j}(\vec{c}),$ (265)

with $ d_{i,j}\geq 0$ for all $ i,j$ and $ p_{i,j}\geq 0$ for all $ i\not= j$.

We basically consider two types of ecosystem models. In the simple NPZ (nutrient-phytoplankton-zooplankton) type models all state variables are based on the same measurable unit such as [mmol N m$ ^{-3}$] for nitrogen-based models. In such NPZ models the reactive terms do only exchange mass between state variables with

$\displaystyle p_{i,j}(\vec{c}) = d_{j,i}(\vec{c}),$    for $\displaystyle i \not= j$    and $\displaystyle \quad p_{i,i}(\vec{c}) = d_{i,i}(\vec{c})=0,$    for $\displaystyle i = j.$ (266)

Neglecting for a moment all transport terms, it is easily seen that this simple type of model is conserving mass:

\begin{displaymath}\begin{array}{l} \displaystyle d_t\left(\sum_{i=1}^I c_i \rig...
...\left(p_{i,i}(\vec{c}) - d_{i,i}(\vec{c})\right)=0. \end{array}\end{displaymath} (267)

The NPZD model (see section [*]) and the Fasham et al. (1990) model discussed in section [*] are such fully conservative models.

In many biogeochemical models most state variables are known to be positive or at least non-negative quantities. For non-negative initial conditions $ c_i(0) \geq 0$ one can easily show by a simple contradiction argument that the condition

$\displaystyle d_{i,j}(\vec{c}) \longrightarrow 0 \;\;$   for$\displaystyle \;\; c_i \longrightarrow 0$ (268)

guarantees that the quantities $ c_i(t) \geq 0$, remain non-negative for all $ t$. A typical example is $ d_{i,j}(\vec{c}) = f(\vec{c}) c_i$ with a non-negative, bounded function $ f$ which might depend on all $ c_i$.

However, for many applications such simple models are too restrictive. Often different spatial references are involved for the state variables, such as the detritus concentration in the water column, measured in [mmol N m$ ^{-3}$] and the fluff layer concentration at the bed, measured in [mmol N m$ ^{-2}$]. Many biogeochemical processes involve more than two substances such as the photosynthesis where different nutrients (e.g. nitrate and phosphorus) are taken up by phytoplankta and oxygen is produced. The ratios between these substances dissipated or produced are usually fixed, in the example of photosynthesis uptake of 16 mmol m$ ^{-3}$ nitrate is connected to an uptake of 1 mmol m$ ^{-3}$ phosphorus and a production of 8.125 mmol m$ ^{-3}$ oxygen.

For state variables which may be negative (such as oxygen concentration which also includes oxygen demand units, all sink and source terms are added up in the production terms $ p_{i,j}$, with a negative sign for the sink terms. For the Neumann et al. (2002) model discussed in sections [*], further deviations from the conservation formulation are introduced since biogeochemical reactions include substances which are not budgeted by the model (mostly because they are assumed to be not limiting). One typical example is nitrogen fixated by blue-green algae which builds up biomass by using atmospheric nitrogen which is later recycled to nitrate. Such non-conservative terms are lumped into the diagonal terms $ p_{i,i}$ and $ d_{i,i}$.

Karsten Bolding 2012-01-24