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% Code to solve the elastic sea level equation following
% Kendall et al., 2005 and Austermann et al., 2015
% J. Austermann 2016
% add paths when run for the first time.
addpath SLFunctions
%% Parameters & Input
% Specify maximum degree to which spherical transformations should be done
maxdeg = 128;
% Some options to choose from
include_rotation = 'y'; % choose between y (for yes) and n (for no)
include_ice_check = 'y'; % choose between y (for yes) and n (for no)
% parameters
rho_ice = 920;
rho_water = 1000;
rho_sed = 2300;
g = 9.80665;
% The following steps help speed up the calculations
% Set up Gauss Legendre grid onto which to interpolate all grids
N = maxdeg; %change size if you want to run it at higher resolution than the maxdeg
[x,w] = GaussQuad(N);
x_GL = acos(x)*180/pi - 90;
lon_GL = linspace(0,360,2*N+1);
lon_GL = lon_GL(1:end-1);
colat = 90 - x_GL;
lon = lon_GL;
[lon_out,lat_out] = meshgrid(lon_GL,x_GL);
% Precompute legendre polynomials
P_lm = cell(N+1,1);
for l=0:N
P_lm{l+1} = legendre(l,x,'norm');
end
% --------------------------------
% ICE
% --------------------------------
% load ice6g
load ice_grid/ice6g_data
ice_in = ice6g;
ice = single(zeros(length(x_GL),length(lon_GL),length(ice_time)));
ice_time_new = zeros(size(ice_time));
ice_lat = [90; ice_lat; -90];
ice_long = [-0.5; ice_long; 360.5];
for i = 1:length(ice_time)
% put onto on Gauss Legendre grid
ice_nointerp = squeeze(ice_in(i,:,:));
% add rows at top and bottom, and right
ice_extended = [zeros(1,length(ice_long)-2); ice_nointerp; ...
ice_nointerp(1,end)*ones(1,length(ice_long)-2)];
ice_extended_2 = [ice_extended(:,end), ice_extended, ice_extended(:,1)];
% interpolate ice masks on Gauss Legendre grid
ice_interp = interp2(ice_long,ice_lat,ice_extended_2,lon_out,lat_out);
ice(:,:,length(ice_time)-i+1) = ice_interp;
ice_time_new(i) = ice_time(length(ice_time)-i+1);
end
% --------------------------------
% TOPOGRAPHY
% --------------------------------
% load preloaded etopo, which includes interpolated fields onto different
% sized Gauss Legendre Grids (hence avoiding interpolating twice)
load topo_SL
% interpolate topography grid onto Gauss Legendre Grid
if N == 64
topo_pres = topo_bed_64 + ice(:,:,end);
elseif N == 128
topo_pres = topo_bed_128 + ice(:,:,end);
elseif N == 256
topo_pres = topo_bed_256 + ice(:,:,end);
elseif N == 512
topo_pres = topo_bed_512 + ice(:,:,end);
elseif N == 1024
topo_pres = topo_bed_1024 + ice(:,:,end);
else
topo_pres = interp2(lon_topo,lat_topo,topo_bed,lon_out,lat_out) + ice(:,:,end);
end
oc_pres = sign_01(topo_pres);
ocpres_lm = sphere_har(oc_pres,maxdeg,N,P_lm);
oc_area = ocpres_lm(1);
% --------------------------------
% SEDIMENT
% --------------------------------
% sediment history
sed = single(zeros(length(x_GL),length(lon_GL),length(ice_time_new)));
%% Set up love number input
% prepare love numbers in suitable format and calculate T_lm and E_lm
% to calculate the fluid case, switch h_el to h_fl, k_el to k_fl and same
% for tidal love numbers
%load /Users/jackyaustermann/Desktop/SLcode/SavedLN/prem.l96C.umVM5.lmVM5.mat
load SavedLN/prem.l96C.ump5.lm5.mat
h_lm = love_lm(h_el, maxdeg);
k_lm = love_lm(k_el, maxdeg);
h_lm_tide = love_lm(h_el_tide,maxdeg);
k_lm_tide = love_lm(k_el_tide,maxdeg);
E_lm = 1 + k_lm - h_lm;
T_lm = get_tlm(maxdeg);
E_lm_T = 1 + k_lm_tide - h_lm_tide;
% calculate betas
beta_l = cell(length(ice_time_new)-1,1);
beta_konly_l = cell(length(ice_time_new)-1,1);
for t_it = 2:length(ice_time_new)
for n = 2:t_it-1
beta = zeros(maxdeg, 1);
for lm = 1:maxdeg
num_mod = mode_found(lm);
beta(lm) = sum((k_amp(lm,1:num_mod) - h_amp(lm,1:num_mod)) ...
./spoles(lm,1:num_mod).* (1 - exp(- spoles(lm,1:num_mod) ...
* (-ice_time_new(t_it) + ice_time_new(n)))));
end
beta_l{t_it-1}(n-1,:) = [0; beta]; % add 0 LN
% for rotation only needed for degree 2
lm = 2;
num_mod = mode_found(lm);
beta_konly_l{t_it-1}(n-1) = sum((k_amp(lm,1:num_mod)) ...
./spoles(lm,1:num_mod).* (1 - exp(- spoles(lm,1:num_mod) ...
* (-ice_time_new(t_it) + ice_time_new(n)))));
end
end
% calculate tidal betas
beta_tide = cell(length(ice_time_new)-1,1);
beta_konly_tide = cell(length(ice_time_new)-1,1);
for t_it = 2:length(ice_time_new)
for n = 2:t_it-1
beta = zeros(maxdeg, 1);
for lm = 1:maxdeg
num_mod = mode_found(lm);
beta(lm) = sum((k_amp_tide(lm,1:num_mod) - h_amp_tide(lm,1:num_mod)) ...
./spoles(lm,1:num_mod).* (1 - exp(- spoles(lm,1:num_mod) ...
* (-ice_time_new(t_it) + ice_time_new(n)))));
end
beta_tide{t_it-1}(n-1,:) = [0; beta]; % add 0 LN
% for rotation only needed for degree 2
lm = 2;
num_mod = mode_found(lm);
beta_konly_tide{t_it-1}(n-1) = sum((k_amp_tide(lm,1:num_mod)) ...
./spoles(lm,1:num_mod).* (1 - exp(- spoles(lm,1:num_mod) ...
* (-ice_time_new(t_it) + ice_time_new(n)))));
end
end
% initiate mapping from l to lm
beta_counter = ones(size(h_lm));
l_it = 1;
for lm_it = 1:length(h_lm)
if lm_it == l_it*(l_it+1)/2
beta_counter(lm_it+1) = beta_counter(lm_it)+1;
l_it = l_it+1;
else
beta_counter(lm_it+1) = beta_counter(lm_it);
end
end
%% Solve sea level equation (after Kendall 2005, Dalca 2013 & Austermann et al. 2015)
tic
k_max = 10; % maximum number of iterations
epsilon = 10^-4; % convergence criterion
topo_it_max = 3; % maximum number of iterations
max_topo_diff = 1; % convergence criterion
% 0 = before
% j = after
% set up initial topography and ocean function
topo_initial = zeros(length(x_GL),length(lon_GL),topo_it_max+1);
topo_initial(:,:,1) = topo_pres - ice(:,:,end) - sed(:,:,end) + ice(:,:,1) + sed(:,:,1); % already includes ice, DT and sediments
% initial topography guess: topography is the same as present at every
% point in time but we account for ice and sediment changes;
% topography is a 3D vector; access topography at time x
% like this topo(:,:,x) [or for plotting squeeze(topo(:,:,x))]
topo = zeros(length(x_GL),length(lon_GL),length(ice_time_new));
for i = 2:length(ice_time_new)
topo(:,:,i) = topo_pres - ice(:,:,end) - sed(:,:,end) + ice(:,:,i) + sed(:,:,1);
end
% initialize
sdelS_lm = zeros(length(ice_time_new),length(h_lm));
ESL = zeros(size(ice_time_new));
ice_corrected = ice;
% initial values for convergence
conv_topo = 'not converged yet';
% TOPOGRAPHY ITERATION
for topo_it = 1:topo_it_max
switch conv_topo
case 'converged!'
case 'not converged yet'
% initialize for each timestep
delL_lm_prev = zeros(1,length(h_lm));
delS_lm_prev = zeros(1,length(h_lm));
TO_lm_prev = zeros(1,length(h_lm));
delLa_lm_prev = zeros(1,length(h_lm));
deli_00_prev = 0;
sdelL_lm = zeros(length(ice_time_new)-1,length(h_lm));
sdelLa_lm = zeros(length(ice_time_new)-1,length(h_lm));
sdelI = zeros(length(ice_time_new)-1,3);
sdelm = zeros(length(ice_time_new)-1,3);
% update new initial topography
topo(:,:,1) = topo_initial(:,:,topo_it);
% remove the corrected ice model and add initial ice model back on
% this needs to be done to calculate the updated corrected ice model
for i = 1:length(ice_time_new)
topo(:,:,i) = topo(:,:,i) - ice_corrected(:,:,i) + ice(:,:,i);
end
% recompute corrected ice model
% do grounded ice check to calculate the corrected ice model
for i = 1:length(ice_time_new)
if include_ice_check == 'y'
% check ice model for floating ice
check1 = sign_01(-topo(:,:,i) + ice(:,:,i));
check2 = sign_01(+topo(:,:,i) - ice(:,:,i)) .* ...
(sign_01(-ice(:,:,i)*rho_ice - (topo(:,:,i) - ice(:,:,i))*rho_water));
ice_corrected(:,:,i) = check1.*ice(:,:,i) + check2.*ice(:,:,i);
else
% if the floating ice check is set to 'n' that don't change the
% ice model
ice_corrected(:,:,i) = ice(:,:,i);
end
end
% update all topographies with the new / corrected ice model
for i = 1:length(ice_time_new)
topo(:,:,i) = topo(:,:,i) - ice(:,:,i) + ice_corrected(:,:,i);
end
% assign topography of time 0 and calculate ocean functions
topo_0 = topo(:,:,1);
oc_0 = sign_01(topo_0);
oc0_lm = sphere_har(oc_0,maxdeg,N,P_lm);
ocj_lm_prev = oc0_lm;
% TIME ITERATION
for t_it = 2:length(ice_time_new) % loop over time
% Assign topography and ocean function of time t_it to the
% index j
topo_j = topo(:,:,t_it);
oc_j = sign_01(topo_j);
ocj_lm = sphere_har(oc_j,maxdeg,N,P_lm);
% calculate topography correction
TO = topo_0.*(oc_j-oc_0);
TO_lm = sphere_har(TO,maxdeg,N,P_lm);
% calculate change in sediments and decompose into spherical harmonics
% set to zero as is but can be set as the change between the sed load.
del_sed = sed(:,:,t_it) - sed(:,:,1);
delSed_lm = sphere_har(del_sed,maxdeg,N,P_lm);
% calculate the change in ice model
del_ice_corrected = ice_corrected(:,:,t_it) - ice_corrected(:,:,1);
deli_lm = sphere_har(del_ice_corrected,maxdeg,N,P_lm);
% calculate the incremental increase in ice volume
sdeli_00 = deli_lm(1) - deli_00_prev;
% initial values for convergence
conv = 'not converged yet';
% SEA LEVEL EQUATION ITERATION
for k = 1:k_max % loop for sea level and topography iteration
switch conv
case 'converged!'
case 'not converged yet'
% set up initial guess for sea level change
if k == 1 && topo_it == 1
% initial guess of sea level change is just to distribute the
% ice over the oceans
% use slightly different initial guess than Kendall
sdelS_lm(t_it,:) = ocj_lm_prev/ocj_lm_prev(1)*...
(-rho_ice/rho_water*sdeli_00 + ...
TO_lm(1)-TO_lm_prev(1)) ...
- TO_lm - TO_lm_prev;
end
delS_lm = delS_lm_prev + sdelS_lm(t_it,:);
% calculate change in loading
% delL is total change in loading
delL_lm = rho_ice*deli_lm + rho_water*delS_lm ...
+ rho_sed*delSed_lm;
% sdelL (small delta L) is incremental change in load -
% relative to last time step
sdelL_lm(t_it-1,:) = delL_lm - delL_lm_prev;
% calculate viscous contribution
% beta contains the viscous love numbers for time t_it,
% row index goes over the time increments, column
% index goes over lm
if t_it == 2
V_lm = zeros(size(T_lm));
else
for lm_it = 1:length(h_lm)
V_lm(lm_it) = beta_l{t_it-1}(:,beta_counter(lm_it))'...
* sdelL_lm(1:t_it-2,lm_it);
end
end
% calculate contribution from rotation
if include_rotation == 'y'
[delLa_lm, sdelI, sdelm] = calc_rot_visc(delL_lm,...
k_el(2),k_el_tide(2),t_it,...
beta_konly_l, beta_konly_tide,...
sdelI, sdelm);
sdelLa_lm(t_it-1,:) = delLa_lm - delLa_lm_prev;
if t_it == 2
V_lm_T = zeros(size(T_lm));
else
for lm_it = 1:6 % don't need to loop over all degrees
V_lm_T(lm_it) = beta_tide{t_it-1}(:,beta_counter(lm_it))'...
* sdelLa_lm(1:t_it-2,lm_it);
end
end
% calculate sea level perturbation
% add ice and sea level and multiply with love numbers
% DT doesn't load!
delSLcurl_lm_fl = E_lm .* T_lm .* delL_lm + T_lm .* V_lm + ...
1/g*E_lm_T.*delLa_lm + 1/g*V_lm_T;
% if don't include rotation
else
delSLcurl_lm_fl = E_lm .* T_lm .* delL_lm + ...
T_lm .* V_lm;
end
% convert to spherical harmonics and subtract terms that are part
% of the topography to get the 'pure' sea level change
delSLcurl_fl = inv_sphere_har(delSLcurl_lm_fl,maxdeg,N,P_lm);
delSLcurl = delSLcurl_fl - del_ice_corrected - del_sed;
% compute and decompose RO
RO = delSLcurl.*oc_j;
RO_lm = sphere_har(RO,maxdeg,N,P_lm);
% calculate eustatic sea level perturbation (delta Phi / g)
delPhi_g = 1/ocj_lm(1) * (- rho_ice/rho_water*deli_lm(1) ...
- RO_lm(1) + TO_lm(1));
sdelS_lm_new = RO_lm + delPhi_g.*ocj_lm - TO_lm ...
- delS_lm_prev;
% calculate convergence criterion chi
chi = abs((sum(abs(sdelS_lm_new)) - sum(abs(sdelS_lm(t_it,:)))) / ...
sum(abs(sdelS_lm(t_it,:))) );
% check convergence against the value epsilon
% If converged, set the variable conv to 'converged!' so that the
% calculation exits the loop. If not converged iterate again.
if chi < epsilon
conv = 'converged!';
disp(['Finished time ' num2str(ice_time_new(t_it))...
'kyr. Number of iterations ' num2str(k) '. delphi is ' num2str(delPhi_g)])
% disp(['Converged after iteration ' num2str(k) '. Chi was ' num2str(chi) '.'])
elseif chi < epsilon && k == k_max;
conv = 'not converged yet';
disp(['Finished time ' num2str(ice_time_new(t_it))...
'kyr. Run has not converged. Chi is ' num2str(chi)])
else
conv = 'not converged yet';
%disp(['Finished iteration ' num2str(k) '. Chi was ' num2str(chi) '.'])
end
% update sea sea surface height
sdelS_lm(t_it,:) = sdelS_lm_new;
end
end
delS_lm_prev = delS_lm;
TO_lm_prev = TO_lm;
delL_lm_prev = delL_lm;
deli_00_prev = deli_lm(1);
ESL(t_it) = -deli_lm(1)/oc_area * rho_ice/rho_water;
if include_rotation == 'y'
delLa_lm_prev = delLa_lm;
end
% calculate overall perturbation of sea level over oceans
% (spatially varying field and constant offset)
delSL = delSLcurl + delPhi_g;
% write in topography for next iteration
topo(:,:,t_it) = - delSL + topo_0;
ocj_lm_prev = ocj_lm;
end
topo_pres_ice_corrected = topo_pres - ice(:,:,end) + ice_corrected(:,:,end);
topo_diff = max(max(abs(topo(:,:,end) - topo_pres_ice_corrected)));
if topo_diff < max_topo_diff
conv_topo = 'converged!';
disp(['Converged!! Number of topo iterations ' num2str(topo_it) ...
'. Topo_diff is ' num2str(topo_diff)])
else
conv_topo = 'not converged yet';
disp(['Not converged. Number of topo iterations ' num2str(topo_it) ...
'. Topo_diff is ' num2str(topo_diff)])
end
end
% update initial topography
topo_initial(:,:,topo_it+1) = topo_pres_ice_corrected - (topo(:,:,end) - topo(:,:,1));
end
toc
% calculate relative sea level (note that topography includes ice and we
% therefore need to subtract it here)
RSL = zeros(size(topo));
for i = 1:length(ice_time_new)
RSL(:,:,i) = (topo(:,:,end) - ice_corrected(:,:,end)) - ...
(topo(:,:,i) - ice_corrected(:,:,i));
end
% calculate ESL relative to present
ESL = ESL - ESL(end);
clear ESL_plot ESL_plot_time
ind = 1;
for i = 1:2:2*length(ESL)-2
ESL_plot(i) = ESL(ind);
ESL_plot(i+1) = ESL(ind);
ESL_plot_time(i) = ice_time_new(ind);
ESL_plot_time(i+1) = ice_time_new(ind+1);
ind = ind+1;
end
%save('ice6g_VM5','ESL_plot','ESL_plot_time','RSL','lon_out','lat_out')
%% Plot results at time
fig_time = 18;
ind = find(ice_time_new==fig_time);
plotSL = squeeze(RSL(:,:,ind));
figure
hold on
pcolor(lon_out,lat_out,plotSL)
%axis([200 330 15 80])
shading flat
contour(lon_out,lat_out,topo_pres,[0 0],'k')
%caxis([-20 20])
colorbar
% if using m_Map
% % plot
% figure
% m_proj('robinson','clongitude',0);
% m_pcolor([lon_out(:,end/2+1:end)-360 lon_out(:,1:end/2)],lat_out,...
% [plotSL(:,end/2+1:end) plotSL(:,1:end/2)])
% m_coast('color',[0 0 0]);
% m_grid('box','fancy','xticklabels',[],'yticklabels',[]);
% shading flat
% colorbar
% colormap(jet)
%% Plot results at point
lon_pt = 90;
lat_pt = 22;
% We only want the sea level change cause by melted ice, so subtract
% del_ice
clear RSL_pt
for i = 1:length(ice_time_new)
RSL_pt(i) = interp2(lon_out,lat_out,squeeze(RSL(:,:,i)),lon_pt,lat_pt);
end
% plot
figure
plot(ice_time_new,RSL_pt)
hold on
plot(ESL_plot_time,ESL_plot,'k')
set(gca,'XDir','Reverse')
xlim([0 26])
legend({'Relative sea level','Eustatic sea level - constant ocean area'} ...
,'Location','NorthWest','FontSize',12)
xlabel('time (ka)')
ylabel('Sea level (m)')
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