-
Notifications
You must be signed in to change notification settings - Fork 0
/
steady_coulomb.m
247 lines (217 loc) · 10.9 KB
/
steady_coulomb.m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
close all
clear all
clc
%% Bed parameters
params.b0 = -100; %bed topo at x=0
params.bx = -1e-3; %linear bed slope
params.sill_min = 2000e3; %sill min x position
params.sill_max = 2100e3; %sill max x position
params.sill_slope = 1e-3; %slope of sill
%% Physical parameters
params.A = 2.9e-25;
params.n = 3;
params.rho_i = 917;
params.rho_w = 1028;
params.g = 9.81;
params.C = 0.2; % base 0.2
params.As = 2.26e-21; % Calculated
params.f = 0.07; % From Kingslake thesis
params.K0 = 10^-24; % From Kingslake thesis
params.L = 3.3e5; % Kingslake thesis
params.year = 3600*24*365;
%% Scaling params (coupled model equations solved in non-dim form)
params.x0 = 100*10^3;
params.h0 = 1000;
params.Q0 = 1;
params.psi0 = params.rho_w*params.g*params.h0/params.x0;
params.M0 = params.Q0/params.x0;
params.m0 = params.Q0*params.psi0/params.L;
params.eps_r = params.m0*params.x0/(params.rho_i*params.Q0);
params.S0 = (params.f*params.rho_w*params.g*params.Q0^2/params.psi0)^(3/8);
params.th0 = params.rho_i*params.S0/params.m0;
params.N0 = (params.K0*params.th0)^(-1/3);
params.delta = params.N0/(params.x0*params.psi0);
%params.u0 = (rho_i*g*params.h0^2/(C*params.N0*params.x0))^n;
params.u0 = params.As*(params.C*params.N0)^params.n;
params.t0 = params.x0/params.u0;
params.a0 = params.h0/params.t0;
params.alpha = 2*params.u0^(1/params.n)/(params.rho_i*params.g*params.h0*(params.x0*params.A)^(1/params.n));
%gamma = As*(C*N0)^n;
params.gamma = (params.C*params.N0*params.x0)/(params.rho_i*params.g*params.h0^2);
params.beta = params.th0/params.t0;
params.r = params.rho_i/params.rho_w;
params.transient = 0;
%% Grid parameters - ice sheet
params.Nx = 3000; %number of grid points - 200
params.Nh = 3000;
fine_grid = linspace(0,1,2*params.Nx);
params.sigma = fine_grid(2:2:end)'; %grid points on velocity (includes GL, not ice divide)
params.sigma_elem = fine_grid(1:2:end)';
params.dsigma = diff(params.sigma_elem); %grid spacing
%% Grid parameters - hydro
params.sigma_h = linspace(0,1,params.Nh)';
params.dsigma_h = diff(params.sigma_h); %grid spacing
%% Establish timings
params.year = 3600*24*365; %number of seconds in a year
params.Nt =50; %number of time steps - normally 150
params.end_year = 1000; %normally 7500
params.dt = params.end_year*params.year/params.Nt;
%% Determine at what points there is coupling
% 1 - coupling on, 0 - coupling off
params.hydro_u_from_ice_u = 1;
params.hydro_psi_from_ice_h = 1;
params.ice_N_from_hydro = 1;
%% Initial "steady state" conditions
params.shear_scale = 1;
Q = ones(params.Nh,1);
N = ones(params.Nh,1);
S = ones(params.Nh,1);
params.S_old = S;
params.M = 1e-5/params.M0; % zero when using schoof bed
params.N_terminus = 0;
params.accum = 1./params.year;
xg = 240e3/params.x0;
hf = (-bed(xg.*params.x0,params)/params.h0)/params.r;
h = 1 - (1-hf).*params.sigma;
u = 0.2*(params.sigma_elem.^(1/3)) + 1e-3; % 0.1 for C = 0.5, 0.3 for C = 0.1-0.4
params.Q_in = 0.01/params.Q0;
params.h_old = h;
params.xg_old = xg;
params.ice_start = 3*params.Nh;
sig_old = params.sigma;
sige_old = params.sigma_elem;
QNShuxg0 = [Q;N;S;h;u;xg];
options = optimoptions('fsolve','Display','iter','SpecifyObjectiveGradient',false,'MaxFunctionEvaluations',1e6,'MaxIterations',1e3);
flf = @(QNShuxg) coulomb_flatbed_highres_steady(QNShuxg,params);
[QNShuxg_init,F,exitflag,output,JAC] = fsolve(flf,QNShuxg0,options);
Q = QNShuxg_init(1:params.Nh);
N = QNShuxg_init(params.Nh+1:2*params.Nh);
S = QNShuxg_init(2*params.Nh+1:3*params.Nh);
h = QNShuxg_init(params.ice_start+1:params.ice_start+ params.Nx);
u = QNShuxg_init(params.ice_start + params.Nx+1:params.ice_start+2*params.Nx);
xg = QNShuxg_init(params.ice_start+2*params.Nx+1);
hf = (-bed(xg.*params.x0,params)/params.h0)/(params.r);
%% Plot terms
plot_terms(QNShuxg_init,params);
figure()
ax1 = subplot(5,1,1);
plot(params.sigma_h.*xg.*params.x0./1000,Q); ylabel('Q');title('Nondimensionalized steady state variables');
ax2 =subplot(5,1,2);
plot(params.sigma_h.*xg.*params.x0./1000,N);ylabel('N');hold on;
ax3 = subplot(5,1,3);
plot(params.sigma_h.*xg.*params.x0./1000,S);ylabel('S');hold on;
ax4 = subplot(5,1,4);
b_h = -bed(xg.*params.sigma_elem.*params.x0,params)./params.h0;
plot(params.sigma_elem.*xg.*params.x0./1000,h-b_h);ylabel('h');hold on;plot(params.sigma_elem.*xg.*params.x0./1000,-b_h,'k')
ax5 = subplot(5,1,5);
plot(params.sigma.*xg.*params.x0./1000,u);hold on;ylabel('u');xlabel('distance from divide, \emph{x} (km)','Interpreter','latex')
linkaxes([ax1,ax2,ax3,ax4,ax5],'x')
%% helper functions
function plot_terms(QNShuxg,params)
sigma_elem = params.sigma_elem;
sigma_h = params.sigma_h;
sigma = params.sigma;
% Q = QNShuxg(1:params.Nx);
% N = QNShuxg(params.Nx+1:2*params.Nx);
% S = QNShuxg(2*params.Nx+1:3*params.Nx);
% h = QNShuxg(3*params.Nx+1:4*params.Nx);
% u = QNShuxg(4*params.Nx+1:5*params.Nx);
% xg = QNShuxg(5*params.Nx+1);
Q = QNShuxg(1:params.Nh);
N = QNShuxg(params.Nh+1:2*params.Nh);
S = QNShuxg(2*params.Nh+1:3*params.Nh);
h = QNShuxg(params.ice_start+1:params.ice_start+ params.Nx);
u = QNShuxg(params.ice_start + params.Nx+1:params.ice_start+2*params.Nx);
xg = QNShuxg(params.ice_start+2*params.Nx+1);
% do ice plots on sigma_elem grid
x_grid_ice = sigma*xg*params.x0;
diff_ice = diff(x_grid_ice);
figure();
%% Plot 1 - ice mass conservation
% Remember that since these are all in steady state, d/dt goes to zero
accum = params.accum; % normalization done in flowline equations
h_interp = interp1(sigma_elem,h.*params.h0,sigma,"linear","extrap");
dhu_dx = gradient((h_interp.*u).*params.u0)./diff_ice(1);
subplot(3,2,1);
plot(x_grid_ice./1000,accum.*ones(size(x_grid_ice)),'DisplayName', 'Accumulation');
hold on;
plot(x_grid_ice./1000,dhu_dx,'--','DisplayName', '$\frac{\partial (hu)}{\partial x}$');
legend('Interpreter','latex');
xlabel('distance from divide, \emph{x} (km)','Interpreter','latex')
title('$\frac{\partial h_g}{\partial t} + \frac{\partial (h_gu)}{\partial x}= a$','Interpreter','latex')
%% Plot 2: ice momentum conservation
N_interp = interp1(sigma_h,N,sigma,"linear","extrap");
shear_stress = params.shear_scale.*params.C.*N_interp(2:end)*params.N0.*(u(2:end).*params.u0./(u(2:end).*params.u0+params.As*(params.C*N_interp(2:end)*params.N0).^params.n)).^(1/params.n);
b = -bed(x_grid_ice,params);
driving_stress = params.rho_i*params.g*h_interp(2:end).*gradient(h_interp(2:end)-b(2:end))./gradient(x_grid_ice(2:end));
shear_and_driving = shear_stress + driving_stress;
dudx = gradient(u(2:end).*params.u0)./gradient(x_grid_ice(2:end));
long_stress = gradient(2*params.A^(-1/params.n)*h_interp(2:end).*abs(dudx).^(1/params.n -1).*dudx)./gradient(x_grid_ice(2:end));
subplot(3,2,2);
plot(x_grid_ice(2:end)./1000,shear_stress,'DisplayName', 'Shear stress');
hold on;
plot(x_grid_ice(2:end)./1000,driving_stress,'DisplayName', 'Driving stress');
plot(x_grid_ice(2:end)./1000,long_stress,'DisplayName', 'Longitudinal stress');
plot(x_grid_ice(2:end)./1000,shear_and_driving,'DisplayName', 'Shear + driving stress');
legend;
xlabel('distance from divide, \emph{x} (km)','Interpreter','latex')
title('$\frac{\partial }{\partial x}\left[ 2\bar{A}^{-1/n}h_g\left|\frac{\partial u}{\partial x}\right|^{1/n-1}\frac{\partial u}{\partial x}\right] - CN(\frac{u}{u+A_sC^nN^n})^{1/n} -\rho_igh_g\frac{\partial (h_g-b)}{\partial x}=0$','Interpreter','latex')
%% Plot 3: Channel Cross sectional area
x_grid_hydro = sigma_h*xg*params.x0;
m_over_rho_i = params.m0.*abs(Q).^3./(params.rho_i.*S.^(8/3)); % from m'=|Q'|^3/S'^8/3
KSN3 = params.K0.*S.*params.S0.*(N.*params.N0).^3;
u_interp = interp1(sigma,u.*params.u0,sigma_h,"linear","extrap");
u_advect = u_interp.*gradient(S.*params.S0)./gradient(x_grid_hydro);
subplot(3,2,3);
plot(x_grid_hydro./1000,m_over_rho_i,'k','DisplayName', 'Melt term');
hold on;
plot(x_grid_hydro./1000,KSN3,'--','DisplayName', '$K_0SN^3$');
plot(x_grid_hydro./1000,u_advect,':','DisplayName', 'Advection term');
plot(x_grid_hydro./1000,KSN3 + u_advect,'x','DisplayName', '$K_0SN^3$ + Advection term');
legend('Interpreter','latex');
xlabel('distance from divide, \emph{x} (km)','Interpreter','latex')
title('$\frac{\partial S}{\partial t_h} = \frac{m}{\rho_i}-K_0SN^3 - u\frac{\partial S}{\partial x}$',Interpreter='latex')
%% Plot 4: Conservation of mass
dQdx = gradient(Q.*params.Q0)./gradient(x_grid_hydro);
m_over_rho_w = m_over_rho_i.*params.rho_i./params.rho_w;
M = ones(size(x_grid_hydro)).*params.M*params.M0;
subplot(3,2,4);
plot(x_grid_hydro./1000,dQdx,'DisplayName', '$\frac{\partial Q}{\partial x}$');
hold on;
plot(x_grid_hydro./1000,m_over_rho_w,'DisplayName', 'Melt term');
plot(x_grid_hydro./1000,M,'DisplayName', 'Supply term');
legend('Interpreter','latex');
xlabel('distance from divide, \emph{x} (km)','Interpreter','latex')
title('$\frac{\partial S}{\partial t_h} + \frac{\partial Q}{\partial x} = \frac{m}{\rho_w}+M$','Interpreter','latex')
%% Plot 5: Momentum conservation
dNdx = gradient(N.*params.N0)./gradient(x_grid_hydro);
f_Q_S = params.f*params.rho_w*params.g.*Q.*params.Q0.*abs(Q.*params.Q0)./(S.*params.S0).^(8/3);
h_interp = interp1(sigma_elem,h,sigma_h,'linear','extrap');
b = -bed(x_grid_hydro,params);
psi = params.rho_w*params.g*gradient(b)./gradient(x_grid_hydro) - gradient(params.rho_i*params.g*h_interp*params.h0)./gradient(x_grid_hydro);
subplot(3,2,5);
plot(x_grid_hydro./1000,psi,'DisplayName', '$\psi$');
hold on;
plot(x_grid_hydro./1000,dNdx,'DisplayName', '$\frac{\partial N}{\partial x}$');
plot(x_grid_hydro./1000,f_Q_S,'DisplayName', '$f\rho_wg\frac{Q|Q|}{S^{8/3}}$');
plot(x_grid_hydro./1000,psi+dNdx,'--','DisplayName', '$\psi+\frac{\partial N}{\partial x}$');
legend('Interpreter','latex');
xlabel('distance from divide, \emph{x} (km)','Interpreter','latex')
title('$\psi+ \frac{\partial N}{\partial x} = f\rho_wg\frac{Q|Q|}{S^{8/3}}$','Interpreter','latex')
%% Plot 6: Energy
melt_term = m_over_rho_i.*params.rho_i*params.L;
flow_through_potential = Q.*params.Q0 .*(psi + dNdx);
subplot(3,2,6);
plot(x_grid_hydro./1000,melt_term,'DisplayName', 'Melt');
hold on;
plot(x_grid_hydro./1000,flow_through_potential,'DisplayName', 'Flow through potential');
legend('Interpreter','latex');
xlabel('distance from divide, \emph{x} (km)','Interpreter','latex')
title('$mL = Q(\psi + \frac{\partial N}{\partial x})$','Interpreter','latex')
figure()
plot(x_grid_hydro./1000,params.rho_w*params.g*gradient(b)./gradient(x_grid_hydro),'DisplayName','$\rho_wg\frac{\partial b}{\partial x}$');
hold on;
plot(x_grid_hydro./1000,gradient(params.rho_i*params.g*h_interp*params.h0)./gradient(x_grid_hydro),'DisplayName','$\rho_ig\frac{\partial h}{\partial x}$')
plot(x_grid_hydro./1000,psi,'DisplayName', '$\psi$');
legend('Interpreter','latex');
end