**3-D PhEcoM (Physical-Ecosystem (biogeochemical) Model) in the polar
and subpolar seas**

We extend the 1-D plankton model of Eslinger et al. (2001) to the
following 3-D *PhEcoM*, whose dynamics can be expressed by a set of
controlling equations as follows:

(For details, please refer to Wang et al. 2003; the flowchart for the
nine compartments is illustrated in Fig. 1)

$$\begin{array}{r}
\frac{\partial D}{\partial t} = - \overrightarrow{V} \cdot \nabla D + \nabla_{h} \cdot \left( K_{h} \cdot \nabla_{h}D \right) \\
 + D(G - R - \text{Rg}) - \Gamma\text{ZS} - \Gamma\text{ZL} - \Gamma\text{ZP} + \frac{\partial(SD)}{\partial z} + K\frac{\partial D}{\partial z^{2}}
\end{array}$$

$$\begin{array}{r}
\frac{\partial F}{\partial t} = - \overrightarrow{V} \cdot \nabla F + \nabla_{h} \cdot \left( K_{h} \cdot \nabla_{h}F \right) \\
 + F(G - R - \text{Rg}) - \Gamma\text{ZS} - \Gamma\text{ZL} - \Gamma\text{ZP} + \frac{\partial(SF)}{\partial z} + K\frac{\partial F}{\partial z^{2}}
\end{array}$$

$$\begin{array}{r}
\frac{\partial\text{ZS}}{\partial t} = - \overrightarrow{V} \cdot \nabla\text{ZS} + \nabla_{h} \cdot \left( K_{h} \cdot \nabla_{h}\text{ZS} \right) \\
 + \text{ZS}(A^{S}(\Gamma + \Gamma)(1 - \text{Ex}) - M) + K\frac{\partial\text{ZS}}{\partial z^{2}}
\end{array}$$

$\begin{array}{r}
\frac{\partial\text{ZL}}{\partial t} = - \overrightarrow{V} \cdot \nabla\text{ZL} + \nabla_{h} \cdot \left( K_{h} \cdot \nabla_{h}\text{ZL} \right) \\
 + \text{ZL}(A(\Gamma + \Gamma)(1 - \text{Ex}) - M) + \frac{\partial(S\text{ZL})}{\partial z} + K\frac{\partial\text{ZL}}{\partial z^{2}}
\end{array}$

$$\begin{array}{r}
\frac{\partial\text{ZP}}{\partial t} = - \overrightarrow{V} \cdot \nabla\text{ZP} + \nabla_{h} \cdot \left( K_{h} \cdot \nabla_{h}\text{ZP} \right) \\
 + \text{ZP}(A(\Gamma + \Gamma)(1 - \text{Ex}) - M) + K\frac{\partial\text{ZP}}{\partial z^{2}}
\end{array}$$

$$\begin{array}{r}
\frac{\partial\text{NO}_{3}}{\partial t} = - \overrightarrow{V} \cdot \nabla\text{NO}_{3} + \nabla_{h} \cdot \left( K_{h} \cdot \nabla_{h}\text{NO}_{3} \right) \\
 - f_{\text{NO}3}(D(G - R) + F(G - R)) + K\frac{\partial\text{NO}_{3}}{\partial z^{2}}
\end{array}$$

$$\begin{array}{r}
\frac{\partial\text{NH}_{4}}{\partial t} = - \overrightarrow{V} \cdot \nabla\text{NH}_{4} + \nabla_{h} \cdot \left( K_{h} \cdot \nabla_{h}\text{NH}_{4} \right) \\
 + \text{ZS} \cdot A(\Gamma + \Gamma)\text{Ex} + \text{ZL} \cdot A(\Gamma^{\text{DL}} + \Gamma)\text{Ex}^{L} \\
 + \text{ZP} \cdot A^{P}(\Gamma^{\text{DP}} + \Gamma^{\text{FP}})\text{Ex}^{P} + D \cdot \text{Rg}^{D} + F \cdot \text{Rg}^{F} + \text{Fec} \cdot \text{Rg}^{\text{Fec}} \\
 - \left( 1 - f_{\text{NO}3} \right)(D(G^{D} - R^{D}) + F(G^{F} - R^{F})) + K\frac{\partial\text{NH}_{4}}{\partial z^{2}}
\end{array}$$

$$\begin{array}{r}
\frac{\partial\text{Si}}{\partial t} = - \overrightarrow{V} \cdot \nabla\text{Si} + \nabla_{h} \cdot \left( K_{h} \cdot \nabla_{h}\text{Si} \right) \\
 - k_{\text{Si}} \cdot D(G - R) + K\frac{\partial\text{Si}}{\partial z^{2}}
\end{array}$$

$$\begin{array}{r}
\frac{\partial\text{Fec}}{\partial t} = - \overrightarrow{V} \cdot \nabla\text{Fec} + \nabla_{h} \cdot \left( K_{h} \cdot \nabla_{h}\text{Fec} \right) \\
 + \text{ZS} \cdot ((1 - A^{S})(\Gamma + \Gamma) + M^{S}) \\
 + \text{ZL} \cdot ((1 - A^{L})(\Gamma^{\text{DL}} + \Gamma) + M^{L}) \\
 + \text{ZP} \cdot ((1 - A^{P})(\Gamma^{\text{DP}} + \Gamma) + M^{P}) - \text{Fec} \cdot \text{Rg}^{\text{Fec}}
\end{array}$$

$$\begin{array}{r}
\frac{\partial\text{Deep}}{\partial t} = - \overrightarrow{V} \cdot \nabla\text{Deep} + \nabla_{h} \cdot \left( K_{h} \cdot \nabla_{h}\text{Deep} \right) \\
 - \frac{\partial(S^{D}D)}{\partial z} - K\frac{\partial D}{\partial z^{2}} - \frac{\partial(S^{F}F)}{\partial z} - K\frac{\partial^{2}F}{\partial z^{2}} - K\frac{\partial^{2}\text{ZS}}{\partial z^{2}} - \frac{\partial(S^{L}\text{ZL})}{\partial z} - K\frac{\partial^{2}\text{ZL}}{\partial z^{2}} - K\frac{\partial^{2}\text{ZP}}{\partial z^{2}}
\end{array}$$

where$\nabla$ stands for three-dimensional derivation, and$\nabla_{h}$
stands for horizontal derivation. Other parameters are explained in the
following table.

**Table: Parameters and values.**

+------------+-------------------------------------------+------------+
| Parameter  | Definition                                | Value      |
+------------+-------------------------------------------+------------+
| T          | Water temperature                         | variable,  |
|            |                                           | deg. C     |
+------------+-------------------------------------------+------------+
| X          | Biomass of X=D, diatoms and X=F,          | variable,  |
|            | flagellates                               | **μ**MN    |
+------------+-------------------------------------------+------------+
| ZY         | Biomass of Y=S, small copepods; Y=L,      | variable,  |
|            | large copepods; and Y=P, other large      | **μ**MN    |
|            | zooplankton                               |            |
+------------+-------------------------------------------+------------+
| NO~3~,     | Nitrate + nitrite, ammonium               | variable,  |
| NH~4~      |                                           | **μ**MN    |
+------------+-------------------------------------------+------------+
| Si         | Silicon                                   | variable,  |
|            |                                           | **μ**MSi   |
+------------+-------------------------------------------+------------+
| Fec        | Fecal material pool, includes natural     | variable,  |
|            | zooplankton morality, regeneration can    | **μ**MN    |
|            | occur from this pool.                     |            |
+------------+-------------------------------------------+------------+
| Deep       | Pool for biomass which has sunk out of    | variable,  |
|            | the model domain. Needed for conservation | **μ**MN    |
|            | of mass. No regeneration from this pool.  |            |
+------------+-------------------------------------------+------------+
| G^X^       | Maximum temperature-dependent             | m~0~^D^    |
|            | phytoplankton X growth rate:              | =0.06h^-1^ |
|            |                                           |            |
|            | $$G^{x} = m_{o}^                          | r          |
|            | {x}e^{r^{x}T}\text{min}(N_{\text{frac}},\ | ^D^=0.0633 |
|            | text{Si}_{\text{frac}},I_{\text{frac}})$$ |            |
|            |                                           | deg^-1^    |
|            | where m${}_{o}^{x}$ is the growth rate at |            |
|            | 0$o$C,                                    | m~0~^F^    |
|            |                                           | =0.06h^-1^ |
|            | r^x^ is the temperature coefficient, and  |            |
|            | N~frac~, Si~frac~ and I~frac~ are         | r          |
|            | unitless ratios expressing nutrient and   | ^F^=0.0633 |
|            |                                           |            |
|            | light limitation.                         | deg^-1^    |
+------------+-------------------------------------------+------------+
| R^X^       | Respiration rate of phytoplankton X,      | R^x^=0     |
|            |                                           | .05m~0~^X^ |
|            | Set to 5% of growth rate                  |            |
|            |                                           | exp(r^X^T) |
+------------+-------------------------------------------+------------+
| Rg^X^      | Mortality and extracellular excretion of  | Rg~0~^D^=  |
|            | phytoplankton X                           | 9.23e${}^{ |
|            |                                           | - 4}$h^-1^ |
|            | $$\text{Rg}^{x} =                         |            |
|            |  \text{Rg}_{o}^{x}e^{\text{rg}^{x_{T}}}$$ | rg^D^=0.03 |
|            |                                           |            |
|            | (from data in Harrison 1980, see Eslinger | deg^-1^    |
|            | and Iverson 2001)                         |            |
|            |                                           | R          |
|            |                                           | g~0~^F^=9. |
|            |                                           | 23e-4h^-1^ |
|            |                                           |            |
|            |                                           | Rg         |
|            |                                           | ^Fec^=0.03 |
|            |                                           |            |
|            |                                           | deg^-1^    |
+------------+-------------------------------------------+------------+
| $\         | Grazing rate of zooplankton Y on          | Γ          |
| Gamma$^XY^ | phytoplankton X                           | ~DS~=0.005 |
|            |                                           |            |
|            | $$\Gamma^{\t                              | h^-1^      |
|            | ext{XY}} = \Gamma_{\text{xy}}(1 - e^{- I_ |            |
|            | {\text{XY}} \cdot (X - X_{\text{oy}})})$$ | I~DS~=     |
|            |                                           | 1.5μMN^-1^ |
|            | (Ivlev, 1945; Magley, 1990)               |            |
|            |                                           | D~0        |
|            |                                           | S~=0.25μMN |
|            |                                           |            |
|            |                                           | Γ          |
|            |                                           | ~DL~=0.009 |
|            |                                           |            |
|            |                                           | h^-1^      |
|            |                                           |            |
|            |                                           | I~DL~=2    |
|            |                                           | .73μMN^-1^ |
|            |                                           |            |
|            |                                           | D~         |
|            |                                           | 0L~=0.5μMN |
|            |                                           |            |
|            |                                           | Γ~DP~      |
|            |                                           | =0.01h^-1^ |
|            |                                           |            |
|            |                                           | I~DP~=2    |
|            |                                           | .73μMN^-1^ |
|            |                                           |            |
|            |                                           | D~         |
|            |                                           | 0P~=0.5μMN |
|            |                                           |            |
|            |                                           | Γ~FS~=     |
|            |                                           | 0.013h^-1^ |
|            |                                           |            |
|            |                                           | I~FS~=2    |
|            |                                           | .73μMN^-1^ |
|            |                                           |            |
|            |                                           | F~0        |
|            |                                           | S~=0.25μMN |
|            |                                           |            |
|            |                                           | Γ~FL~=     |
|            |                                           | 0.003h^-1^ |
|            |                                           |            |
|            |                                           | I~FL~=2    |
|            |                                           | .73μMN^-1^ |
|            |                                           |            |
|            |                                           | F~         |
|            |                                           | 0L~=0.5μMN |
|            |                                           |            |
|            |                                           | Γ~FP~      |
|            |                                           | =0.01h^-1^ |
|            |                                           |            |
|            |                                           | I~FP~=2    |
|            |                                           | .73μMN^-1^ |
|            |                                           |            |
|            |                                           | F~         |
|            |                                           | 0P~=0.5μMN |
+------------+-------------------------------------------+------------+
| S^X^ ,     | Variable sinking rate for phytoplankton X | S^D^~m     |
| S^ZL^      | or zooplankton Y                          | ax~=4m/day |
|            |                                           |            |
|            | $$S^{x}(\text{NO}_{3                      | S^F^~ma    |
|            | }) = S_{\text{max}} \cdot (1 - \text{tanh | x~=.5m/day |
|            | }(S_{\text{KNO}3} \cdot \text{NO}_{3}))$$ |            |
|            |                                           | tanh(      |
|            | (Eslinger and Iverson 2001)               | S~KNO3•~K~ |
|            |                                           | sno3~)=5.0 |
|            | S ^ZL^ varies with time as described in   |            |
|            | the text.                                 |            |
+------------+-------------------------------------------+------------+
| K          | Vertical mixing coefficient. Only written | Unused     |
|            | this way for clarity. Mixing actually     |            |
|            | performed using Froude number criteria    |            |
|            | (see Eslinger and Iverson, 2001,          |            |
|            | Eslinger, 1990; Thompson, 1976).          |            |
+------------+-------------------------------------------+------------+
| A^Y^       | Assimilation ratio of grazed              | A^S^=      |
|            | phytoplankton                             | A^L^=      |
|            |                                           | A^P^=0.3   |
+------------+-------------------------------------------+------------+
| Ex^Y^      | Egestion ratio of grazed phytoplankton    | Ex^S^=     |
|            |                                           | Ex^L^=     |
|            |                                           | Ex^P^=0.7  |
+------------+-------------------------------------------+------------+
| M^Y^       | Natural mortality rate of zooplankton Y   | M^S^=      |
|            |                                           | M^L^=      |
|            |                                           | M^P^       |
|            |                                           | =0.06d^-1^ |
+------------+-------------------------------------------+------------+
| N~frac~    | $$\begin{array}{r}                        | KS^D^~N    |
|            | N_{\text{frac}}                           | O3~=2.5μMN |
|            | = \frac{\text{NO}_{3}}{\text{KS}_{\text{N |            |
|            | O}3} + \text{NO}_{3}} \cdot \text{exp}\le | KS^D^~N    |
|            | ft( - \psi \cdot \text{NH}_{4} \right) \\ | H4~=2.5μMN |
|            | \lbrack(\frac{\text{NH}_{4}}{\te          |            |
|            | xt{KS}_{\text{NH}4} + \text{NH}_{4}} + \f | ψ^D^=1.    |
|            | rac{\text{NO}_{3}}{\text{KS}_{\text{NO}3} | 462μMN^-1^ |
|            |  + \text{NO}_{3}} \cdot \text{exp}\left(  |            |
|            | - \psi \cdot \text{NH}_{4} \right)\rbrack | KS^F^~N    |
|            | \end{array}$$                             | O3~=0.6μMN |
|            |                                           |            |
|            | (after Wroblewski, 1977)                  | KS^F^~N    |
|            |                                           | H4~=0.6μMN |
|            |                                           |            |
|            |                                           | ψ^F^=1.    |
|            |                                           | 462μMN^-1^ |
+------------+-------------------------------------------+------------+
| Si~frac~   | $$\text{Si}_{\text{frac}} = \frac{\text{  | KS~S       |
|            | Si}}{\text{KS}_{\text{Si}} + \text{Si}}$$ | i~=3.0μMSi |
|            |                                           |            |
|            | (after Dugdale, 1967)                     |            |
+------------+-------------------------------------------+------------+
| I~frac~    | $$I_{\text{frac                           | A^D^/P~S~  |
|            | }} = \left( 1 - \text{exp}\left( - \text{ | ^B^=0.0536 |
|            | aI}P_{s}^{B} \right) \right) \cdot \text{ |            |
|            | exp}\left( - \text{bI}P_{s}^{B} \right)$$ | b          |
|            |                                           | ^D^/P~S~^B |
|            | (after Platt et al. 1980)                 | ^=1.795e-3 |
|            |                                           |            |
|            |                                           | A^F^/P~S~  |
|            |                                           | ^B^=0.0536 |
|            |                                           |            |
|            |                                           | B          |
|            |                                           | ^F^/P~S~^B |
|            |                                           | ^=1.795e-3 |
|            |                                           |            |
|            |                                           | (Wm^-2^)   |
+------------+-------------------------------------------+------------+
| f~NO3~     | Fraction of phytoplankton growth due to   | No unit    |
|            | nitrate uptake over that due to ammonium  |            |
|            | uptake:                                   |            |
|            |                                           |            |
|            | $$\begin{array}{r}                        |            |
|            | a = \frac{\text{NO}3}{\text{KS}_{\        |            |
|            | text{NO}3} + \text{NO}3} \cdot \text{exp} |            |
|            | \left( - \psi \cdot \text{NH}4 \right) \\ |            |
|            | b = \frac{\text{NH}4                      |            |
|            | }{\text{KS}_{\text{NH}4} + \text{NH}4} \\ |            |
|            | f_{\text{NO}3} = \frac{a}{a + b}          |            |
|            | \end{array}$$                             |            |
+------------+-------------------------------------------+------------+
| k~si~      | Silicon to nitrogen mass ratio for diatom | 1.104      |
|            |                                           | (moles     |
|            |                                           | Si)/(moles |
|            |                                           | N)         |
+------------+-------------------------------------------+------------+

**The biochemical model for the DMS dynamics**

The DMS (***d***i-***m***ethyl ***s***ulfide), based on Jodwalis et al.
(2000), can be further extended to the 3D advection-diffusion equation:

$$\frac{\partial\text{DMS}}{\partial t} = - \overrightarrow{V} \cdot \nabla\text{DMS} + \nabla_{h} \cdot \left( k_{h}\nabla_{h} \right) + \frac{\partial}{\partial z}\left( k_{v}\frac{\partial\text{DMS}}{\partial z} \right) + P_{1} + P_{2} + P_{3} - L_{1} - L_{2} - L_{3}$$

where the DMS production is written by:

P~1~---Metabolic release^b^:

> P~1~=DMS metabolically release (mol h^-1^) = 0.000417 h^-1^\*
>
> \[ratio~diatoms~ x mg of diatom Chl.*a*) +(ratio~flagellates~ x mg of
> flagellate Chl.*a*)\]

P~2~---Senescence^c^:

P~2~=DMS released at senescence = 0.00292\*

(ratio~diatoms~ x mg of diatom Chl.*a*) + 0.0108 h^-1^
(ratio~flagellates~ x mg of

> flagellate Chl.*a*)

P~3~---Grazing^b^:

P~3~=DMS released when grazed = 0.01375 h^-1^

\[ratio~diatoms~ x mg of diatom Chl.*a*) + (ratio~flagellates~ x mg of
flagellate Chl.*a*)\]

ratio~diatoms~^a^= 64 x 10^-7^ mol DMSP/mg Chl.*a*

ratio~flagellates~^a^ = 420 x 10^-7^ mol DMSP/mg Chl.*a*

and the DMS loss can be expressed by

L~1~---Sea-to-air flux^d^:

L~1~ = k~LM~ \[DMS\]~sea~

$K_{\text{LM}} = \begin{array}{r}
\left\{ 0\text{.}\text{17}\text{uR}^{0\text{.}\text{65}},u \leq 3\text{.}6\frac{m}{s} \middle| \right.\ \left\{ (2\text{.}\text{85}u - 9\text{.}\text{65})R^{0\text{.}\text{65}},3\text{.}6 < u \leq \text{13}\frac{m}{s} \middle| \right.\ 
\end{array}$

k~LM~ is the transfer velocity in units of (cm h^-1^)

u = windspeed in m s^-1^

T = temperature in ^o^C

R = Sc~CO[2]{.smallcaps}~(20^o^C)/Sc~DMS~(T).

R is the ratio of the Schmidt number of CO~2~ at 20^o^C (660) to the
Schmidt number of DMS at the surface seawater temperature,

Sc~DMS~(T) = 2674.0 -- 147.12T + 3.726T^2^ -- 0.038T^3^

L~2~---Photooxidation^e^:

L~2~ = K\[DMS\]~sea~ e^-kz^

K= 0.09 hr^-1^

k= attenuation coefficient of light through seawater

z = depth

L${}_{3}$---Microbial consumption^f^:

L${}_{3}$ =0.000417 nM h^-1^ or 0.000417 umol m^-3^ h^-1^

References:

[Wang, J.,]{.underline} C. Deal, Z. Wan, M. Jin, N. Tanaka and M. Ikeda,
2003. *User's Guide for a*

*Ph*ysical-*Eco*system *M*odel (*PhEcoM*) in the Subpolar and Polar
Oceans.

International Arctic Research Center-Frontier Research System for Global
Change,

Tech. Rep. 02-02, 69 pp.

Jin, M. C.J. Deal, [J. Wang]{.underline}, K.-H. Shin, N. Tanaka, T.
Whitledge, S. H. Lee, R.R.

Gradinger, 2006. Controls of the landfast ice-ocean ecosystem offshore
Barrow,

Alaska. *Annals of Glaciology (iin press*)

Jin, M., C.J. Deal, [J. Wang,]{.underline} N. Tanaka, and M. Ikeda,
2006. Vertical mixing effects on the

phytoplankton bloom in the southeastern Bering Sea mid-shelf. *J.
Geophys. Res*., 111,

C03002, doi:10.1029/2005JC002994.

Deal, C.J., M. Jin, [J. Wang]{.underline}**,** N. Tanaka, M. Ikeda, T.
Whitledge and T. Rho, 2006. Dynamics

of plankton and nutrients in the Bering Sea: An ecosystem model study
with special

focus on the coccolithophore bloom and water column nitrification
(submitted to

*Prog. Oceanogr*.)