# 8.6.1. THSICE: The Thermodynamic Sea Ice Package¶

Important note: This document has been written by Stephanie Dutkiewicz and describes an earlier implementation of the sea-ice package. This needs to be updated to reflect the recent changes (JMC).

This thermodynamic ice model is based on the 3-layer model by Winton (2000). and the energy-conserving LANL CICE model (Bitz and Lipscomb, 1999). The model considers two equally thick ice layers; the upper layer has a variable specific heat resulting from brine pockets, the lower layer has a fixed heat capacity. A zero heat capacity snow layer lies above the ice. Heat fluxes at the top and bottom surfaces are used to calculate the change in ice and snow layer thickness. Grid cells of the ocean model are either fully covered in ice or are open water. There is a provision to parametrize ice fraction (and leads) in this package. Modifications are discussed in small font following the subroutine descriptions.

## 8.6.1.1. Key parameters and Routines¶

The ice model is called from thermodynamics.F, subroutine ice_forcing.F is called in place of external_forcing_surf.F.

In ice_forcing.F, we calculate the freezing potential of the ocean model surface layer of water:

${\bf frzmlt} = (T_f - SST) \frac{c_{\rm sw} \rho_{\rm sw} \Delta z}{\Delta t}$

where $$c_{\rm sw}$$ is seawater heat capacity, $$\rho_{\rm sw}$$ is the seawater density, $$\Delta z$$ is the ocean model upper layer thickness and $$\Delta t$$ is the model (tracer) timestep. The freezing temperature, $$T_f=\mu S$$ is a function of the salinity.

1. Provided there is no ice present and frzmlt is less than 0, the surface tendencies of wind, heat and freshwater are calculated as usual (ie. as in external_forcing_surf.F).

2. If there is ice present in the grid cell we call the main ice model routine ice_therm.F (see below). Output from this routine gives net heat and freshwater flux affecting the top of the ocean.

Subroutine ice_forcing.F uses these values to find the sea surface tendencies in grid cells. When there is ice present, the surface stress tendencies are set to zero; the ice model is purely thermodynamic and the effect of ice motion on the sea-surface is not examined.

Relaxation of surface $$T$$ and $$S$$ is only allowed equatorward of relaxlat (see DATA.ICE below), and no relaxation is allowed under the ice at any latitude.

(Note that there is provision for allowing grid cells to have both open water and seaice; if compact is between 0 and 1)

### 8.6.1.1.1. subroutine ICE_FREEZE¶

This routine is called from thermodynamics.F after the new temperature calculation, calc_gt.F, but before calc_gs.F. In ice_freeze.F, any ocean upper layer grid cell with no ice cover, but with temperature below freezing, $$T_f=\mu S$$ has ice initialized. We calculate frzmlt from all the grid cells in the water column that have a temperature less than freezing. In this routine, any water below the surface that is below freezing is set to $$T_f$$. A call to ice_start.F is made if frzmlt $$>0$$, and salinity tendancy is updated for brine release.

(There is a provision for fractional ice: In the case where the grid cell has less ice coverage than icemaskmax we allow ice_start.F to be called)

### 8.6.1.1.2. subroutine ICE_START¶

The energy available from freezing the sea surface is brought into this routine as esurp. The enthalpy of the 2 layers of any new ice is calculated as:

\begin{split}\begin{aligned} q_1 & = -c_{i}*T_f + L_i \nonumber \\ q_2 & = -c_{f}T_{\rm mlt}+ c_{i}(T_{\rm mlt}-T{f}) + L_i(1-\frac{T_{\rm mlt}}{T_f}) \nonumber\end{aligned}\end{split}

where $$c_f$$ is specific heat of liquid fresh water, $$c_i$$ is the specific heat of fresh ice, $$L_i$$ is latent heat of freezing, $$\rho_i$$ is density of ice and $$T_{\rm mlt}$$ is melting temperature of ice with salinity of 1. The height of a new layer of ice is

$h_{i \rm new} = \frac{{\bf esurp} \Delta t}{qi_{0av}}$

where $$qi_{0av}=-\frac{\rho_i}{2} (q_1+q_2)$$.

The surface skin temperature $$T_s$$ and ice temperatures $$T_1$$, $$T_2$$ and the sea surface temperature are set at $$T_f$$.

(There is provision for fractional ice: new ice is formed over open water; the first freezing in the cell must have a height of himin0; this determines the ice fraction compact. If there is already ice in the grid cell, the new ice must have the same height and the new ice fraction is

$i_f=(1-\hat{i_f}) \frac{h_{i \rm new}}{h_i}$

where $$\hat{i_f}$$ is ice fraction from previous timestep and $$h_i$$ is current ice height. Snow is redistributed over the new ice fraction. The ice fraction is not allowed to become larger than iceMaskmax and if the ice height is above hihig then freezing energy comes from the full grid cell, ice growth does not occur under orginal ice due to freezing water.)

### 8.6.1.1.3. subroutine ICE_THERM¶

The main subroutine of this package is ice_therm.F where the ice temperatures are calculated and the changes in ice and snow thicknesses are determined. Output provides the net heat and fresh water fluxes that force the top layer of the ocean model.

If the current ice height is less than himin then the ice layer is set to zero and the ocean model upper layer temperature is allowed to drop lower than its freezing temperature; and atmospheric fluxes are allowed to effect the grid cell. If the ice height is greater than himin we proceed with the ice model calculation.

We follow the procedure of Winton (1999) – see equations 3 to 21 – to calculate the surface and internal ice temperatures. The surface temperature is found from the balance of the flux at the surface $$F_s$$, the shortwave heat flux absorbed by the ice, fswint, and the upward conduction of heat through the snow and/or ice, $$F_u$$. We linearize $$F_s$$ about the surface temperature, $$\hat{T_s}$$, at the previous timestep (where $$\hat{ }$$ indicates the value at the previous timestep):

$F_s (T_s) = F_s(\hat{T_s}) + \frac{\partial F_s(\hat{T_s)}}{\partial T_s} (T_s-\hat{T_s})$

where,

$F_s = F_{\rm SH}+F_{\rm LH}+F_{LW \downarrow}+F_{LW \uparrow} + (1- \alpha) F_{\rm SW}$

and

$\frac{d F_s}{dT} = \frac{d F_{\rm SH}}{dT} + \frac{d F_{\rm LH}}{dT} +\frac{d F_{\rm LW \uparrow}}{dT}.$

$$F_s$$ and $$\frac{d F_s}{dT}$$ are currently calculated from the BULKF package described separately, but could also be provided by an atmospheric model. The surface albedo is calculated from the ice height and/or surface temperature (see below, srf_albedo.F) and the shortwave flux absorbed in the ice is

${\bf fswint} = (1-e^{\kappa_i h_i})(1-\alpha) F_{SW}$

where $$\kappa_i$$ is bulk extinction coefficient.

The conductive flux to the surface is

$F_u=K_{1/2}(T_1-T_s)$

where $$K_{1/2}$$ is the effective conductive coupling of the snow-ice layer between the surface and the mid-point of the upper layer of ice $$K_{1/2}=\frac{4 K_i K_s}{K_s h_i + 4 K_i h_s}$$, $$K_i$$ and $$K_s$$ are constant thermal conductivities of seaice and snow.

From the above equations we can develop a system of equations to find the skin surface temperature, $$T_s$$ and the two ice layer temperatures (see Winton, 1999, for details). We solve these equations iteratively until the change in $$T_s$$ is small. When the surface temperature is greater then the melting temperature of the surface, the temperatures are recalculated setting $$T_s$$ to 0. The enthalpy of the ice layers are calculated in order to keep track of the energy in the ice model. Enthalpy is defined, here, as the energy required to melt a unit mass of seaice with temperature $$T$$. For the upper layer (1) with brine pockets and the lower fresh layer (2):

\begin{split}\begin{aligned} q_1 & = - c_f T_f + c_i (T_f-T)+ L_{i}(1-\frac{T_f}{T}) \nonumber \\ q_2 & = -c_i T+L_i \nonumber\end{aligned}\end{split}

where $$c_f$$ is specific heat of liquid fresh water, $$c_i$$ is the specific heat of fresh ice, and $$L_i$$ is latent heat of melting fresh ice.

From the new ice temperatures, we can calculate the energy flux at the surface available for melting (if $$T_s$$=0) and the energy at the ocean-ice interface for either melting or freezing.

\begin{split}\begin{aligned} E_{\rm top} & = & (F_s- K_{1/2}(T_s-T_1) ) \Delta t \nonumber \\ E_{\rm bot} &= & (\frac{4K_i(T_2-T_f)}{h_i}-F_b) \Delta t \nonumber\end{aligned}\end{split}

where $$F_b$$ is the heat flux at the ice bottom due to the sea surface temperature variations from freezing. If $$T_{\rm SST}$$ is above freezing, $$F_b=c_{\rm sw} \rho_{\rm sw} \gamma (T_{\rm SST}-T_f)u^{*}$$, $$\gamma$$ is the heat transfer coefficient and $$u^{*}=QQ$$ is frictional velocity between ice and water. If $$T_{\rm SST}$$ is below freezing, $$F_b=(T_f - T_{\rm SST})c_f \rho_f \Delta z /\Delta t$$ and set $$T_{\rm SST}$$ to $$T_f$$. We also include the energy from lower layers that drop below freezing, and set those layers to $$T_f$$.

If $$E_{\rm top}>0$$ we melt snow from the surface, if all the snow is melted and there is energy left, we melt the ice. If the ice is all gone and there is still energy left, we apply the left over energy to heating the ocean model upper layer (See Winton, 1999, equations 27-29). Similarly if $$E_{\rm bot}>0$$ we melt ice from the bottom. If all the ice is melted, the snow is melted (with energy from the ocean model upper layer if necessary). If $$E_{\rm bot}<0$$ we grow ice at the bottom

$\Delta h_i = \frac{-E_{\rm bot}}{(q_{\rm bot} \rho_i)}$

where $$q_{\rm bot}=-c_{i} T_f + L_i$$ is the enthalpy of the new ice, The enthalpy of the second ice layer, $$q_2$$ needs to be modified:

$q_2 = \frac{ \hat{h_i}/2 \hat{q_2} + \Delta h_i q_{\rm bot} } {\hat{h_i}/{2}+\Delta h_i}$

If there is a ice layer and the overlying air temperature is below 0$$^o$$C then any precipitation, $$P$$ joins the snow layer:

$\Delta h_s = -P \frac{\rho_f}{\rho_s} \Delta t,$

$$\rho_f$$ and $$\rho_s$$ are the fresh water and snow densities. Any evaporation, similarly, removes snow or ice from the surface. We also calculate the snow age here, in case it is needed for the surface albedo calculation (see srf_albedo.F below).

For practical reasons we limit the ice growth to hilim and snow is limited to hslim. We converts any ice and/or snow above these limits back to water, maintaining the salt balance. Note however, that heat is not conserved in this conversion; sea surface temperatures below the ice are not recalculated.

If the snow/ice interface is below the waterline, snow is converted to ice (see Winton, 1999, equations 35 and 36). The subroutine new_layers_winton.F, described below, repartitions the ice into equal thickness layers while conserving energy.

The subroutine ice_therm.F now calculates the heat and fresh water fluxes affecting the ocean model surface layer. The heat flux:

$q_{\rm net}= {\bf fswocn} - F_{b} - \frac{{\bf esurp}}{\Delta t}$

is composed of the shortwave flux that has passed through the ice layer and is absorbed by the water, fswocn$$=QQ$$, the ocean flux to the ice $$F_b$$, and the surplus energy left over from the melting, esurp. The fresh water flux is determined from the amount of fresh water and salt in the ice/snow system before and after the timestep.

(There is a provision for fractional ice: If ice height is above hihig then all energy from freezing at sea surface is used only in the open water aparts of the cell (ie. $$F_b$$ will only have the conduction term). The melt energy is partitioned by frac_energy between melting ice height and ice extent. However, once ice height drops below himon0 then all energy melts ice extent.)

### 8.6.1.1.4. subroutine SFC_ALBEDO¶

The routine ice_therm.F calls this routine to determine the surface albedo. There are two calculations provided here:

1. from LANL CICE model

$\alpha = f_s \alpha_s + (1-f_s) (\alpha_{i_{\min}} + (\alpha_{i_{\max}}- \alpha_{i_{\min}}) (1-e^{-h_i/h_{\alpha}}))$

where $$f_s$$ is 1 if there is snow, 0 if not; the snow albedo, $$\alpha_s$$ has two values depending on whether $$T_s<0$$ or not; $$\alpha_{i_{\min}}$$ and $$\alpha_{i_{\max}}$$ are ice albedos for thin melting ice, and thick bare ice respectively, and $$h_{\alpha}$$ is a scale height.

2. From GISS model (Hansen et al 1983)

$\alpha = \alpha_i e^{-h_s/h_a} + \alpha_s (1-e^{-h_s/h_a})$

where $$\alpha_i$$ is a constant albedo for bare ice, $$h_a$$ is a scale height and $$\alpha_s$$ is a variable snow albedo.

$\alpha_s = \alpha_1 + \alpha_2 e^{-\lambda_a a_s}$

where $$\alpha_1$$ is a constant, $$\alpha_2$$ depends on $$T_s$$, $$a_s$$ is the snow age, and $$\lambda_a$$ is a scale frequency. The snow age is calculated in ice_therm.F and is given in equation 41 in Hansen et al (1983).

### 8.6.1.1.5. subroutine NEW_LAYERS_WINTON¶

The subroutine new_layers_winton.F repartitions the ice into equal thickness layers while conserving energy. We pass to this subroutine, the ice layer enthalpies after melting/growth and the new height of the ice layers. The ending layer height should be half the sum of the new ice heights from ice_therm.F. The enthalpies of the ice layers are adjusted accordingly to maintain total energy in the ice model. If layer 2 height is greater than layer 1 height then layer 2 gives ice to layer 1 and:

$q_1=f_1 \hat{q_1} + (1-f1) \hat{q_2}$

where $$f_1$$ is the fraction of the new to old upper layer heights. $$T_1$$ will therefore also have changed. Similarly for when ice layer height 2 is less than layer 1 height, except here we need to to be careful that the new $$T_2$$ does not fall below the melting temperature.

### 8.6.1.1.6. Initializing subroutines¶

ice_init.F: Set ice variables to zero, or reads in pickup information from pickup.ic (which was written out in checkpoint.F)

### 8.6.1.1.7. Diagnostic subroutines¶

ice_ave.F: Keeps track of means of the ice variables

ice_diags.F: Finds averages and writes out diagnostics

### 8.6.1.1.8. Common Blocks¶

ICE.h: Ice Varibles, also relaxlat and startIceModel

ICE_DIAGS.h: matrices for diagnostics: averages of fields from ice_diags.F

BULKF_ICE_CONSTANTS.h (in BULKF package): all the parameters need by the ice model

### 8.6.1.1.9. Input file DATA.ICE¶

Here we need to set StartIceModel: which is 1 if the model starts from no ice; and 0 if there is a pickup file with the ice matrices (pickup.ic) which is read in ice_init.F and written out in checkpoint.F. The parameter relaxlat defines the latitude poleward of which there is no relaxing of surface $$T$$ or $$S$$ to observations. This avoids the relaxation forcing the ice model at these high latitudes.

(Note: hicemin is set to 0 here. If the provision for allowing grid cells to have both open water and seaice is ever implemented, this would be greater than 0)

## 8.6.1.2. Important Notes¶

1. heat fluxes have different signs in the ocean and ice models.

2. StartIceModel must be changed in data.ice: 1 (if starting from no ice), 0 (if using pickup.ic file).

## 8.6.1.3. THSICE Diagnostics¶

------------------------------------------------------------------------
<-Name->|Levs|<-parsing code->|<--  Units   -->|<- Tile (max=80c)
------------------------------------------------------------------------
SI_Fract|  1 |SM P    M1      |0-1             |Sea-Ice fraction  [0-1]
SI_Thick|  1 |SM PC197M1      |m               |Sea-Ice thickness (area weighted average)
SI_SnowH|  1 |SM PC197M1      |m               |Snow thickness over Sea-Ice (area weighted)
SI_Tsrf |  1 |SM  C197M1      |degC            |Surface Temperature over Sea-Ice (area weighted)
SI_Tice1|  1 |SM  C197M1      |degC            |Sea-Ice Temperature, 1srt layer (area weighted)
SI_Tice2|  1 |SM  C197M1      |degC            |Sea-Ice Temperature, 2nd  layer (area weighted)
SI_Qice1|  1 |SM  C198M1      |J/kg            |Sea-Ice enthalpy, 1srt layer (mass weighted)
SI_Qice2|  1 |SM  C198M1      |J/kg            |Sea-Ice enthalpy, 2nd  layer (mass weighted)
SIalbedo|  1 |SM PC197M1      |0-1             |Sea-Ice Albedo [0-1] (area weighted average)
SIsnwAge|  1 |SM P    M1      |s               |snow age over Sea-Ice
SIsnwPrc|  1 |SM  C197M1      |kg/m^2/s        |snow precip. (+=dw) over Sea-Ice (area weighted)
SIflxAtm|  1 |SM      M1      |W/m^2           |net heat flux from the Atmosphere (+=dw)
SIfrwAtm|  1 |SM      M1      |kg/m^2/s        |fresh-water flux to the Atmosphere (+=up)
SIflx2oc|  1 |SM      M1      |W/m^2           |heat flux out of the ocean (+=up)
SIfrw2oc|  1 |SM      M1      |m/s             |fresh-water flux out of the ocean (+=up)
SIsaltFx|  1 |SM      M1      |(g/kg).kg/m^2   |salt flux out of the ocean (+=up)
SItOcMxL|  1 |SM      M1      |degC            |ocean mixed layer temperature
SIsOcMxL|  1 |SM P    M1      |g/kg            |ocean mixed layer salinity


## 8.6.1.5. Experiments and tutorials that use thsice¶

• Global atmosphere experiment in aim.5l_cs verification directory, input from input.thsice directory.

• Global ocean experiment in global_ocean.cs32x15 verification directory, input from input.thsice directory.