Sea-Ice Model - Revision history

Robertson at 13:01, 10 August 2015

2015-08-10T13:01:16Z

← Older revision		Revision as of 13:01, 10 August 2015
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	<math display="block">		<math display="block">
	T_2 & = T_1 + \gamma \left[ \frac{L }{ C_{pw}} + T_1 \left( 1		T_2 = T_1 + \gamma \left[ \frac{L }{ C_{pw}} + T_1 \left( 1
	- \frac{C_{pi} }{ C_{pw}} \right) \right]		- \frac{C_{pi} }{ C_{pw}} \right) \right]
	</math>		</math>
	<math display="block">		<math display="block">
	S_2 & = S_1 (1 + \gamma) \,.		S_2 = S_1 (1 + \gamma) \,.
	</math>		</math>

Robertson at 13:00, 10 August 2015

2015-08-10T13:00:13Z

← Older revision		Revision as of 13:00, 10 August 2015
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	leads to:		leads to:

	<math display="block">~~\begin{align}~~		<math display="block">
	T_2 & = T_1 + \gamma \left[ \frac{L }{ C_{pw}} + T_1 \left( 1		T_2 & = T_1 + \gamma \left[ \frac{L }{ C_{pw}} + T_1 \left( 1
	- \frac{C_{pi} }{ C_{pw}} \right) \right] \\		- \frac{C_{pi} }{ C_{pw}} \right) \right]
	S_2 & = S_1 (1 + \gamma) \,. ~~\end{align}~~		</math>
			<math display="block">
			S_2 & = S_1 (1 + \gamma) \,.
	</math>		</math>

Robertson at 14:45, 5 August 2015

2015-08-05T14:45:49Z

← Older revision		Revision as of 14:45, 5 August 2015
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	\frac{d}{ dt}(A h_i u)		\frac{d}{ dt}(A h_i u)
	&= A h_i fv - A h_i g \frac{\partial \zeta_w }{ \partial x} +		&= A h_i fv - A h_i g \frac{\partial \zeta_w }{ \partial x} +
	\frac{A}{\rho_i}(\tau_a^x + \tau_w^x) + \frac{1}{\rho_i}{\cal F}_x		\frac{A}{\rho_i}(\tau_a^x + \tau_w^x) + \frac{1}{\rho_i}{\cal F}_x \\
	\frac{d}{ d t}(A h_i v)		\frac{d}{ d t}(A h_i v)
	&= - A h_i fu - A h_i g \frac{\partial \zeta_w }{ \partial y} +		&= - A h_i fu - A h_i g \frac{\partial \zeta_w }{ \partial y} +
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	<math display="block">\begin{align}		<math display="block">\begin{align}
	C_{S_z} &= \frac{u_\tau }{ P_{rt} k^{-1}\ln (-z/z_0) + B_S}		C_{S_z} &= \frac{u_\tau }{ P_{rt} k^{-1}\ln (-z/z_0) + B_S} \\
	B_S &= b \left(\frac{z_0 u_\tau }{ \nu} \right) ^{1/2} Sc^{2/3} \end{align}		B_S &= b \left(\frac{z_0 u_\tau }{ \nu} \right) ^{1/2} Sc^{2/3} \end{align}
	</math>		</math>

Robertson at 14:40, 5 August 2015

2015-08-05T14:40:14Z

Show changes

Kate: /* Dynamics */

2014-07-29T15:03:19Z

Dynamics

← Older revision		Revision as of 15:03, 29 July 2014
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	air and water stress, and internal ice stress:		air and water stress, and internal ice stress:

	$$M {~~du \over~~ dt} = ~~M f v~~ - M g {\partial \zeta_w ~~\over~~ \partial x} + \tau_a^x + \tau_w^x + {\cal F}_x$$		$$ \frac{d}{ dt}(A h_i u)
	$$M {~~dv \over dt~~} = - ~~M f u~~ - M g {\partial \zeta_w ~~\over~~ \partial y} + \tau_a^y + \tau_w^y + {\cal F}_y$$		= A h_i fv - A h_i g \frac{\partial \zeta_w }{ \partial x} +
			\frac{A}{\rho_i}(\tau_a^x + \tau_w^x) + \frac{1}{\rho_i}{\cal F}_x$$
			$$\frac{d}{ d t}(A h_i v)
			= - A h_i fu - A h_i g \frac{\partial \zeta_w }{ \partial y} +
			\frac{A}{\rho_i}(\tau_a^y + \tau_w^y) + \frac{1}{\rho_i}{\cal F}_y.$$

	In this model, we neglect the nonlinear advection terms as well as		In this model, we neglect the nonlinear advection terms as well as

Kate: /* Thermodynamics */

2014-07-28T20:08:19Z

Thermodynamics

← Older revision		Revision as of 20:08, 28 July 2014
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	$$ \eqalign{		$$ \eqalign{
	\frac{D A h_i}{Dt} &= \frac{\rho_o}{\rho_i} \left[ A (W_{io} - W_{ai}) + (1-A) W_{ao} + W_{fr} \right] \cr		\frac{D A h_i}{Dt} &= \frac{\rho_o}{\rho_i} \left[ A (W_{io} - W_{ai}) + (1-A) W_{ao} + W_{fr} \right] \cr
	\frac{D A}{Dt} &= \frac{\rho_o A}{\rho_i h_i} \left[ \Phi (1-A) w_{ao} + (1-A) W_{fr} \right] \qquad \qquad 0 \leq A \leq 1 } $$		\frac{D A}{Dt} &= \frac{\rho_o}{\rho_i h_i} \left[ \Phi (1-A) w_{ao} + (1-A) W_{fr} \right] \qquad \qquad 0 \leq A \leq 1 } $$

	The term $Ah_i$ is the "effective thickness", a measure of the ice volume. Its evolution equation is simply quantifying the change in the amount of ice. The ice concentration equation is more interesting in that		The term $Ah_i$ is the "effective thickness", a measure of the ice volume. Its evolution equation is simply quantifying the change in the amount of ice. The ice concentration equation is more interesting in that

Kate: /* Dynamics */ Added nonlinear ice strength

2014-06-30T18:38:25Z

Dynamics: Added nonlinear ice strength

← Older revision		Revision as of 18:38, 30 June 2014
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	$$\dot \epsilon_{ij} \equiv {1 \over 2} \left( {\partial u_i \over \partial x_j} + {\partial u_j \over \partial x_i} \right)$$		$$\dot \epsilon_{ij} \equiv {1 \over 2} \left( {\partial u_i \over \partial x_j} + {\partial u_j \over \partial x_i} \right)$$

	~~$$P = P^* A h_i e^{-C(1-A)}$$~~
	where the nonlinear viscosities are given by		where the nonlinear viscosities are given by
	$$\zeta = { P \over 2 \left[ (\epsilon^2_{11} +		$$\zeta = { P \over 2 \left[ (\epsilon^2_{11} +
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	$$\eta = { \zeta \over e^2 }.$$		$$\eta = { \zeta \over e^2 }.$$

			Hibler's ice strength is given by:

			$$P = P^* A h_i e^{-C(1-A)}$$

			while [[Bibliography#Overland_1988 \| Overland and Pease (1988)]]: advocate a nonlinear strength:

			$$P = P^* A h_i^2 e^{-C(1-A)}$$

			in which $P^*$ now depends on the grid spacing. Both options are available in ROMS.

	We would also like to have an explicit model that can be solved		We would also like to have an explicit model that can be solved

Kate: /* Frazil ice formation */

2012-03-27T02:22:44Z

Frazil ice formation

← Older revision		Revision as of 02:22, 27 March 2012
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	==Frazil ice formation==		==Frazil ice formation==
	<wikitex>		<wikitex>
	Following [[Bibliography#Steele_89 \|Steele ''et al.'' (1989)]], we check to see if any		Following [[Bibliography#Steele_89 \|Steele ''et al.'' (1989)]], we check to see if any of the ocean temperatures are below freezing at the end of each timestep. If so, frazil ice is formed, changing the local temperature and salinity. The ice that forms is assumed to instantly float up to the surface and add to the ice layer there. We balance the mass, heat, and salt before and after the ice is formed:
	of the ocean temperatures are below freezing at the end of each
	timestep. If so, frazil ice is ~~assumed to form~~, changing the local
	temperature and salinity. The ice that forms is assumed to
	instantly float up to the surface and add to the ice layer there.
	We ~~assume balances in~~ the mass, heat, and salt before and after the ice
	is formed:
	$$ \eqalign{		$$ \eqalign{
	m_{w_1} & = m_{w_2} + m_i \cr		m_{w_1} & = m_{w_2} + m_i \cr

Kate: /* Ocean surface boundary conditions */

2012-03-27T02:20:51Z

Ocean surface boundary conditions

← Older revision		Revision as of 02:20, 27 March 2012
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	\|-		\|-
	\| $b$		\| $b$
	\| 3		\| 3.14
	\| factor		\| factor
	\|-		\|-
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	\| 0.4		\| 0.4
	\| von Karman's constant		\| von Karman's constant
			\|-
			\| $K_m$
			\|
			\| vertical viscosity of seawater
	\|-		\|-
	\| $\nu$		\| $\nu$
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	\|		\|
	\| surface salinity		\| surface salinity
			\|-
			\| $S$
			\|
			\| internal ocean salinity
			\|-
			\| $Sc$
			\| 2432
			\| molecular Schmidt number
	\|-		\|-
	\| $\tau_{io}$		\| $\tau_{io}$
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	\|		\|
	\| stress on the ocean from the wind		\| stress on the ocean from the wind
			\|-
			\| $T_0$
			\|
			\| surface ocean temperature
	\|-		\|-
	\| $T$		\| $T$
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	\|}		\|}

			surface salinity ($T_0 = mS_0$) in the presense of ice. We also have $T$ and $S$ at the uppermost computed ocean point ${1 \over 2} dz$ below the surface. In order to solve for $T_0$ and $S_0$, we assume a [[Bibliography#Yaglom_1974 \| Yaglom and Kader, 1974]] logarithmic boundary layer. The upper ocean heat flux is:
	~~The surface ocean is assumed to be at the freezing temperature for the~~ surface salinity ($T_0 = mS$) ~~where we use~~ the ~~salinity from~~ the uppermost ~~model~~ point at $~~z=- \frac~~{1}{ 2}~~\Delta z~~$. ~~From this~~, we ~~can obtain~~ a ~~vertical temperature gradient for the~~ upper ocean ~~to use in the~~ heat flux ~~formula~~:
	$$		$$
	\frac{F_T }{ \rho_o ~~C_po~~} = -C_{T_z} (T_0 - T) \qquad z \rightarrow 0		\frac{F_T }{ \rho_o C_{po}} = -C_{T_z} (T_0 - T) \qquad z \rightarrow 0
	$$		$$
	where		where
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	B_T = b \left(\frac{z_0 u_\tau }{ \nu} \right) ^{1/2} Pr^{2/3}		B_T = b \left(\frac{z_0 u_\tau }{ \nu} \right) ^{1/2} Pr^{2/3}
	$$		$$

	~~Once~~ we have a the ~~value~~ for ~~$F_T$, we can use it to find~~ the ~~ice~~		Likewise, we have the following equation for the surface salt flux:
	~~growth rates~~:		$$
	$$ ~~\eqalign{~~		F_S = -C_{S_z} (S_0 - S) \qquad z \rightarrow 0
	~~W_{io}~~ = ~~\frac{1 }~~{ ~~\rho_o L_o~~} (~~Q_{io}~~ - ~~F_T~~) \cr
	~~W_{ao} = \frac{1 }{~~ \~~rho_o L_o} (Q_{ao} - F_T)}~~
	$$		$$
	where		where
	$$		$$
	~~L_o~~ \~~equiv~~ \~~left[ E~~(~~T_0,1~~) ~~- E~~(~~T_1,r_1)~~ \right]		C_{S_z} = \frac{u_\tau }{ P_{rt} k^{-1}\ln (-z/z_0) + B_S}
			$$
			$$
			B_S = b \left(\frac{z_0 u_\tau }{ \nu} \right) ^{1/2} Sc^{2/3}
	$$		$$

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	W_o & \equiv A W_{io} + (1-A) W_{ao} }		W_o & \equiv A W_{io} + (1-A) W_{ao} }
	$$		$$

			[[Bibliography#MellorGL_1989a \| Mellor and Kantha]] describe solving simultaneously for the five unknowns $W_o$, $T_0$, $S_0$, $F_T$ and $F_S$. Instead, we use the old value of $T_0$ to find $W_{io}$ and therefore $W_o$. Using the new value of $W_{io}$, solve for a new value of $S_0$ and then find the new $T_0$ as the freezing temperature for that salinity:
			$$ \eqalign{
			W_{io} &= {1 \over L_o} \left[ {Q_{io} \over \rho_o} + C_{po}
			C_{T_z} (T_o - T) \right] \cr
			S_0 &= { C_{S_z} S + (W_{ai}H(-W_{ai}) - W_{io})) S_i \over
			C_{S_z} + W_{ai}H(-W_{ai}) + W_{ro} - W_{io} } }
			$$
			Note that the term $W_{ai}H(-W_{ai})$ is the contribution of melting only, not refreezing of melt ponds.

	</wikitex>		</wikitex>

	==Frazil ice formation==		==Frazil ice formation==
	<wikitex>		<wikitex>

Kate: /* Thermodynamics */

2012-03-27T02:00:27Z

Thermodynamics

← Older revision		Revision as of 02:00, 27 March 2012
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	it provides the partitioning between ice melt/growth on the sides vs. on the top and bottom. The parameter $\Phi$ controls this and has differing values for ice melt and retreat. In principle, most of the ice growth is assumed to happen at the base of the ice while rather more of the melt happens on the sides of the ice due to warming of the water in the leads.		it provides the partitioning between ice melt/growth on the sides vs. on the top and bottom. The parameter $\Phi$ controls this and has differing values for ice melt and retreat. In principle, most of the ice growth is assumed to happen at the base of the ice while rather more of the melt happens on the sides of the ice due to warming of the water in the leads.

	The heat fluxes through the ice are based on a simple one layer [[Bibliography#SemtnerAJ_1976a \| Semtner (1976)]] type model with snow on top. The temperature is assumed to be linear within the snow and within the ice. The ice contains brine pockets for a total ice salinity of 5. The surface ocean temperature and salinity is half a $dz$ below the surface. The water right below the surface is assumed to be at the freezing temperature; a logarithmic boundary layer is computed having the temperature and salinity matched at freezing.		The heat fluxes through the ice are based on a simple one layer [[Bibliography#SemtnerAJ_1976a \| Semtner (1976)]] type model with snow on top. The temperature is assumed to be linear within the snow and within the ice. The ice contains brine pockets for a total ice salinity of 3.2 or the surface ocean salinity, whichever is less. The surface ocean temperature and salinity is half a $dz$ below the surface. The water right below the surface is assumed to be at the freezing temperature; a logarithmic boundary layer is computed having the temperature and salinity matched at freezing.

			Similarly, there is a snow equation:
			$$
			{D A h_s \over D t} = \left[ A (W_{s} - W_{sm}) \right]
			$$
			where $W_s$ and $W_{sm}$ are the snowfall and snow melt rates, respectively, in units of equivalent water. Also in the model is the depth of the melt ponds in spring, $h_{mp}$ which can be up to 0.1 m, after which melting ice contributes to $W_{ro}$. The melt ponds are not part of the thermal conductivity computations. They could contribute to the albedo computation, but that has been largely commented out.

	The ice model thermodynamic variables:		The ice model thermodynamic variables:
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	\| density of ice		\| density of ice
	\|-		\|-
	\| $S_i$ = 5 PSU		\| $S_i$ = 3.2 PSU
	\| salinity of the ice		\| salinity of the ice
	\|-		\|-
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	\| $W_{ro}$		\| $W_{ro}$
	\| rate of run-off of surface melt water		\| rate of run-off of surface melt water
			\|-
			\| $W_{s}$
			\| snowfall rate
			\|-
			\| $W_{sm}$
			\| snow melt rate
	\|-		\|-
	\|}		\|}
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	(1 - \alpha_s) SW\!\!\downarrow + \epsilon_s \sigma (T_3+273)^4		(1 - \alpha_s) SW\!\!\downarrow + \epsilon_s \sigma (T_3+273)^4
	$$		$$
	The formulae for sensible heat, latent heat, and ~~incoming~~ longwave ~~and shortwave radiations~~ are the same as in [[Bibliography#ParkinsonCL_1979a \|Parkinson and Washington (1979)]] and are shown in [[Radiant Heat Fluxes]]. The sensible heat is a function of $T_3$, as is the heat flux through the snow $Q_s$. Setting $Q_{ai} = Q_s$, we can solve for $T_3$ by setting $T_3^{n+1} = T_3^n + \Delta T_3$ and linearizing in $\Delta T_3$~~. The temperature $T_3$ is found by an iterative solution of the surface heat flux balance (using the~~		The incoming shortwave and longwave radiations are assumed to come from an atmospheric model. The formulae for sensible heat, latent heat, and outgoing longwave radiation are the same as in [[Bibliography#ParkinsonCL_1979a \|Parkinson and Washington (1979)]] and are shown in [[Radiant Heat Fluxes]]. The sensible heat is a function of $T_3$, as is the heat flux through the snow $Q_s$. Setting $Q_{ai} = Q_s$, we can solve for $T_3$ by setting $T_3^{n+1} = T_3^n + \Delta T_3$ and linearizing in $\Delta T_3$. As in Parkinson and Washington, if $T_3$ is found to be above the melting temperature, it is set to $T_{\rm melt}$ and the extra energy goes into melting the snow or ice:
	~~previous value of $T_1$ in equation \ref{qsnow})~~. As in Parkinson and Washington, if $T_3$ is found to be above the melting temperature, it is set to $T_{\rm melt}$ and the extra energy goes into melting the snow or ice:
	$$ \eqalign{		$$ \eqalign{
	W_{ai} & = \frac{Q_{ai} - Q_{i2} }{ \rho_o L_3} \cr		W_{ai} & = \frac{Q_{ai} - Q_{i2} }{ \rho_o L_3} \cr
	L_3 & \equiv \left[ E(T_3,1) - E(T_1, R_1) \right]		L_3 & \equiv \left[ E(T_3,1) - E(T_1, R_1) \right]
	}$$		}$$
	Note that $L_3 = (1-r)L_i$ plus a small sensible heat correction. We ~~are not storing~~ water on the surface in melt pools~~, so~~ everything melted at the surface is assumed to flow into the ocean ($W_{ro~~} = W_{ai~~}$).		Note that $L_3 = (1-r)L_i$ plus a small sensible heat correction. We can store water on the surface in melt pools to a fixed depth---everything extra melted at the surface is assumed to flow into the ocean as $W_{ro}$. The melt ponds however do not contribute to the heat flux computation.

	Inside the ice there are brine pockets in which there is salt water at the ''in situ'' freezing temperature. It is assumed that the ice has a uniform overall salinity of $S_i$ and that the freezing temperature is a linear function of salinity. The brine fraction $r$ is given by		Inside the ice there are brine pockets in which there is salt water at the ''in situ'' freezing temperature. It is assumed that the ice has a uniform overall salinity of $S_i$ and that the freezing temperature is a linear function of salinity. The brine fraction $r$ is given by