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% Reference: "How Model Complexity Influences Sea Ice Stability",

% T.J.W. Wagner & I. Eisenman, J Clim (2015)

%

% This script numerically solves the system described by eqns (2), (8) and

% (9) in the article above (henceforth WE15).

% For computational convenience a diffusive 'ghost layer' is invoked, as

% described in WE15, Appendix A. This allows us to solve the system using

% an Implicit Euler scheme on the ghost layer (which effectively solves the

% diffusion equation) and a Forward Euler scheme for the evolution of the

% surface enthalpy. For further detailed documentation to go with this

% script, see WE15_NumericIntegration.pdf.

%

% The present script uses the default parameter values of WE15 (Table 1 and

% Section 2d) with no climate forcing (F=0) and the final part of the code

% produces the plot corresponding essentially to Figure 2 in WE15.

%

% The default configuration here runs a simulation for 30 years at 1000

% timesteps/year and a spatial resolution of 100 gridboxes, equally spaced

% between equator and pole.

%

% Till Wagner & Ian Eisenman, Mar 15

%

% Minor bug fix, Jan 2022: in eq A1 S(:,i)->S(:,i+1)

% (and accordingly repeated 1st column at end of S array)

%--------------------------------------------------------------------------

function [tfin, Efin] = WE15_default

%%Model parameters (WE15, Table 1 and Section 2d) -------------------------

D  = 0.6;     %diffusivity for heat transport (W m^-2 K^-1)

A  = 193;     %OLR when T = T_m (W m^-2)

B  = 2.1;     %OLR temperature dependence (W m^-2 K^-1)

cw = 9.8;     %ocean mixed layer heat capacity (W yr m^-2 K^-1)

S0 = 420;     %insolation at equator  (W m^-2)

S1 = 338;     %insolation seasonal dependence (W m^-2)

S2 = 240;     %insolation spatial dependence (W m^-2)

a0 = 0.7;     %ice-free co-albedo at equator

a2 = 0.1;     %ice=free co-albedo spatial dependence

ai = 0.4;     %co-albedo where there is sea ice

Fb = 4;       %heat flux from ocean below (W m^-2)

k  = 2;       %sea ice thermal conductivity (W m^-2 K^-1)

Lf = 9.5;     %sea ice latent heat of fusion (W yr m^-3)

% Tm = 0;     %melting temp., not included in the eqns below

F  = 0;       %radiative forcing (W m^-2)

cg = 0.01*cw; %ghost layer heat capacity(W yr m^-2 K^-1)

tau = 1e-5;   %ghost layer coupling timescale (yr)

%%The default run in WE15, Fig 2 uses the time-stepping parameters: -------

% n=400; % # of evenly spaced latitudinal gridboxes (equator to pole)

% nt=1e3; % # of timesteps per year (approx lower limit of stability)

% dur=200; % # of years for the whole run

%%For a quicker computation, use the parameters: --------------------------

n  = 100;

nt = 1e3;

dur= 30;

dt = 1/nt;

%%Spatial Grid ------------------------------------------------------------

dx = 1/n;     %grid box width

x = (dx/2:dx:1-dx/2)';  %native grid

%%Diffusion Operator (WE15, Appendix A) -----------------------------------

xb = (dx:dx:1.0-dx)';

lambda=D/dx^2*(1-xb.^2); L1=[0; -lambda]; L2=[-lambda; 0]; L3=-L1-L2;

diffop = - diag(L3) - diag(L2(1:n-1),1) - diag(L1(2:n),-1);

%%Definitions for implicit scheme on Tg

cg_tau = cg/tau;

dt_tau = dt/tau;

dc = dt_tau*cg_tau;

kappa = (1+dt_tau)*eye(n)-dt*diffop/cg;

%%Seasonal forcing (WE15 eq.3)

ty = dt/2:dt:1-dt/2;

S=repmat(S0-S2*x.^2,[1,nt])-repmat(S1*cos(2*pi*ty),[n,1]).*repmat(x,[1,nt]);

S=[S S(:,1)];

%%Further definitions

M = B+cg_tau;

aw= a0-a2*x.^2;   %open water albedo

kLf = k*Lf;

%%Initial conditions ------------------------------------------------------

T = 7.5+20*(1-2*x.^2);

Tg = T; E = cw*T;

%%Set up output arrays, saving 100 timesteps/year

vec100 = 1:nt/100:nt*dur;

E100 = zeros(n,dur*100); T100 = E100;

p = 0; m = 0;

%%Integration (see WE15_NumericIntegration.pdf)----------------------------

% Loop over Years ---------------------------------------------------------

for years = 1:dur

% Loop within One Year-------------------------------------------------

for i = 1:nt

%store 100 timesteps per year

m = m+1;

if (p+1)*nt/100 == m

p = p+1;

E100(:,p) = E;

T100(:,p) = T;

end

% forcing

alpha = aw.*(E>0) + ai*(E<0);    % WE15, eq.4

C =alpha.*S(:,i)+cg_tau*Tg-A+F;

% surface temperature

T0 =  C./(M-kLf./E);                 %WE15, eq.A3

T = E/cw.*(E>=0)+T0.*(E<0).*(T0<0);  %WE15, eq.9

% Forward Euler on E

E = E+dt*(C-M*T+Fb);                 %WE15, eq.A2

% Implicit Euler on Tg

Tg = (kappa-diag(dc./(M-kLf./E).*(T0<0).*(E<0)))\ ...

(Tg + (dt_tau*(E/cw.*(E>=0)+(ai*S(:,i+1) ...

-A+F)./(M-kLf./E).*(T0<0).*(E<0))));        %WE15, eq.A1

end

yrs = sprintf('year %d complete',years); disp(yrs)

end

% -------------------------------------------------------------------------

%%output only converged, final year

tfin = linspace(0,1,100);

Efin = E100(:,end-99:end);

Tfin = T100(:,end-99:end);

% -------------------------------------------------------------------------

%WE15, Figure 2: Default Steady State Climatology -------------------------

% -------------------------------------------------------------------------

set(0,'defaulttextinterpreter','latex')

winter=26;    %time of coldest <T>

summer=76;    %time of warmest <T>

%%compute seasonal ice edge

xi = zeros(1,100);

for j = 1:length(tfin)

Ej = Efin(:,j);

if isempty(find(Ej<0,1))==0

xi(j) = x(find(Ej<0,1,'first'));

else

xi(j) = max(x);

end

end

fig = figure(2); clf

% set(fig, 'Position', [2561 361 1920 984]);

%%plot the enthalpy (Fig 2a)

subplot(1,4,1)

clevs = [-40:20:0 50:50:300];

contourf(tfin,x,Efin,clevs)

%%plot ice edge on E

hold on

plot(tfin,xi,'k')

colorbar

%%alternatively, use

% cbarf(Efin,clevs);  %makes nice discrete colorbar

xlabel('$t$ (final year)')

ylabel('$x$')

title('$E$ (J/m$^2$)')

% plot the temperature (Fig 2b)

clevs = -30:5:30;

subplot(1,4,2)

contourf(tfin,x,Tfin,clevs)

% plot ice edge on T

hold on

plot(tfin,xi,'k')

%%plot T=0 contour (the region between ice edge and T=0 contour is the

%%region of summer ice surface melt)

contour(tfin,x,Tfin,[0,0],'r')

colorbar

% cbarf(Tfin,clevs);

xlabel('$t$ (final year)')

ylabel('$x$')

title('$T$ ($^\circ$C)')

%%plot the ice thickness (Fig 2c)

hfin = -(Efin/Lf.*(Efin<0));

subplot(1,4,3)

clevs = 0.0001:.5:4;

contourf(tfin,x,hfin,clevs)

%%plot ice edge on h

hold on

contour(tfin,x,hfin,[0,0])

plot([tfin(winter) tfin(winter)], [0 max(x)],'k')

plot([tfin(summer) tfin(summer)], [0 max(x)],'k--')

xlabel('$t$ (final year)')

ylabel('$x$')

title('$h$ (m)')

colorbar

% cbarf(hfin,round(clevs,1));

%%plot temperature profiles (Fig 2d)

subplot(4,4,4)

plot(x,Tfin(:,summer),'k--')

hold on

plot(x,Tfin(:,winter),'k')

plot([0 1], [0 0],'k')

xlabel('$x$')

ylabel('$T$ ($^\circ$C)')

legend('summer','winter','location','southwest')

%%plot ice thickness profiles (Fig 2e)

subplot(4,4,8)

plot(x,hfin(:,summer),'k--')

hold on

plot(x,hfin(:,winter),'k')

plot([0 1], [0 0],'k')

xlim([0.7 1])

xlabel('$x$')

ylabel('$h$ (m)')

%%plot seasonal thickness cycle at pole (Fig 2f)

subplot(4,4,12)

plot(tfin,hfin(end,:),'k')

xlabel('$t$ (final year)')

ylabel('$h_p$ (m)')

ylim([2 1.1*max(hfin(end,:))])

%%plot ice edge seasonal cycle (Fig 2g)

subplot(4,4,16)

plot(tfin,xideg,'k-')

ylim([0 90])

xlabel('$t$ (final year)')

ylabel('$\theta_i$ (deg)')

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