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profiling.py

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+import numpy as np
+import matplotlib.pyplot as plt
+
+
+##DEFINING CONSTANTS
+H = 584-48 #height of ice sheet, minus for dead ice
+h = 217-48 #step height, minus for dead ice
+fb = 0.29 #bottom sliding coefficient
+He = H - h*(1-fb) #effective height
+lam = 0.33/(3600*24*365) #accumulation rate
+Q = 48*10**-3 #geothermanl heat  flux
+K = 2.43 #thermal conductivity
+Ts = 253.95 #surface temp
+t = 3600*24*30  #time step, one month
+kappa = 44.5/(3600*24*365) #thermal diffusivity
+delz = 1 #spatial step, 1 metre
+
+
+
+##STEADYSTATE INITIALISATION
+n_steady = int((H + 48/delz)) #number of model runs
+z_steady = np.linspace(0, n_steady-1, n_steady) #height array
+w_steady = np.zeros([n_steady, 1]) #vertical velocity array
+coeffs_steady = np.zeros([n_steady, n_steady]) #coefficient matrix M
+funcs_steady = np.zeros([n_steady, 1]) #f vector, where zeroth element is bedrock and final element is surface
+
+
+
+##STEADYSTATE MODEL
+#define vertical velocity with the step-model
+for i in range(0, n_steady-1):
+    if z_steady[i] > h + 48:
+        w_steady[i] = -(lam/He) * (z_steady[i] - 48 - (h*(1 - fb)))
+    if 48 < z_steady[i] < h + 48:
+        w_steady[i] = -fb*lam*(z_steady[i] - 48)/He
+    if 0 < z_steady[i] < 48:
+        w_steady[i] = 0
+
+#generate the coefficient matrix
+for i in range(0, n_steady):
+
+    if i == 0:
+        coeffs_steady[i, i] = -1 #b1
+        coeffs_steady[i, i+1] = 1 #c1
+        funcs_steady[i] = -Q*delz/K #f1
+
+    if i == n_steady-1:
+        coeffs_steady[i, i] = 1 #bn
+        coeffs_steady[i, i-1] = 0 #an
+        funcs_steady[i] = Ts #fn
+
+    elif i != 0 and i != n_steady-1:
+        coeffs_steady[i, i] = -2
+        coeffs_steady[i, i-1] = 1 + ((w_steady[i]*delz)/(2*kappa))
+        coeffs_steady[i, i+1] = 1 - ((w_steady[i]*delz)/(2*kappa))
+
+
+
+temps_steady = np.dot(np.linalg.inv(coeffs_steady),funcs_steady) #steady state temperature array
+
+plt.figure()
+plt.plot(temps_steady-273.15, z_steady) #minus for conversion to Celsius
+plt.xlabel('Temperature (C)')
+plt.ylabel('Height above bedrock (m)')
+plt.title('Steady-state stepmodel')
+
+
+satellite = np.loadtxt(r'C:\Users\elloi\Documents\Uni\Bachelor\Projekt\corrected.txt', delimiter=',') #importing satellite data
+
+
+##NON STEADY STATE INITIALISATION
+Lam = lam*917 #for use in density calculation
+n = int(584/delz) #number of model runs
+num_points = 403 #number of time steps in satellite data
+g  = 9.81 #gravity
+
+z = np.linspace(0, n-1, n) #as before
+w = np.zeros([n])
+coeffs = np.zeros([n,n])
+funcs = np.zeros([n])
+
+##VERTICAL VELOCITY
+
+for i in range(0,n-1):
+    if z[i] > h + 48:
+        w[i] = -(lam/He) * ((z[i] - 48) - (h*(1 - fb)))
+    if 48 < z[i] < h + 48:
+        w[i] = -fb*lam*(z[i] - 48)/He
+    if 0 < z[i] < 48: #if dead ice
+        w[i] = 0
+w[-1] = -lam
+
+
+##DENSITY FUNCTION
+
+d = z[::-1] #finds z reversed, for depths rather than heights
+d_crit = 7.97 #critical depth
+d_co = 57.02 #closeoff depth
+rho_0 = 0.391084 #initial density
+rho_crit = 0.497782 #critical density
+rho_co = 0.820 #close-off density
+lh = 0.42 #accumulation rate
+R = 8.314 #gas constant
+K0 = 11 * np.exp(-10160/(R*Ts)) #Arrhenius type numbers
+K1 = 575 * np.exp(-21400/(R*Ts))
+rho_ice = 0.917 #density of ice
+
+
+rho = np.zeros([n]) #density array, as a function of depth
+
+Z1 = np.zeros([n])#empty array for Z1 func
+Z0 = np.zeros([n]) #empty array for Z0 func
+
+for i in range(0,n):
+
+    if d[i] == 0:
+        rho[i] = (rho_0)
+
+    if 0 < d[i] < d_crit: #critical density and above
+        Z0[i] = np.exp((rho_ice*K0*d[i]) + np.log(rho_0/(rho_ice - rho_0)))
+        rho[i] = ((rho_ice * Z0[i])/(1 + Z0[i]))
+
+    if d_crit < d[i] < d_co: #between critical and closeoff density
+        Z1[i] = np.exp( (rho_ice*K1*(d[i]-d_crit)*(lh**-0.5)) + np.log(0.55/(rho_ice - 0.55)) )
+        rho[i] =((rho_ice * Z1[i])/(1 + Z1[i]))
+
+    if d_co < d[i]:  #below close off depth, constant density
+        rho[i] = rho_ice
+
+##DENSITY DEPENDENT PARAMS
+
+rho_z = rho[::-1] #finding rho as function of z
+
+L = [] #overlying load
+K_ice = [] #thermal conductivity of ice
+c = [] #specific heat capacity using steady state guess
+K_firn = [] #thermal  conductivity of firn
+kappa_firn = [] #thermal diffusivity of firn
+kappa_func = [] #d kappa / d rho
+
+for i in range(0,n):
+    L.append(z[i] * rho_z[i])
+
+    K_ice.append(9.828 * np.exp(-0.0057 * temps_steady[i]))
+
+    c.append(152.5 + (7.122 * temps_steady[i]))
+
+    K_firn.append(K_ice[i] * ((rho[i] / rho_ice)**(1 - (rho[i] / (2 * rho_ice)))))
+
+    kappa_firn.append(K_ice[i] / (c[i] * rho_ice * 1000) * ((rho[i] / rho_ice)**(1 - (rho[i] / (2 * rho_ice)))))
+
+    kappa_func.append(kappa_firn[i] * ((1 / (rho_z[i] * 1000)) - \
+        (1 / (2 * rho_ice * 1000))) * (1 + np.log((rho_z[i]) / (rho_ice))))
+
+
+rho_func = [] #d rho / dz
+for i in range(0,n-1):
+    rho_func.append((rho_z[i+1]-rho_z[i])/delz)
+
+
+
+##GENERATING PROFILE
+
+T_list = [ [] for i in range(num_points+1) ]
+T_list[0] = temps_steady #first element of T = steadystate guess
+
+temps = [ [] for i in range(num_points+1) ] #list of profiles - each profile from a new model run
+temps[0] = temps_steady #steady state guess as the first profile
+
+for j in range(1,num_points): #looping over sat points
+
+    for i in range(0,n): #looping over height steps
+
+        if i == 0: #first row
+
+            coeffs[i,i] = -1 #b1
+            coeffs[i,i+1] = 1 #c1
+            funcs[i] = -Q * delz/ K_ice[-1] #f1
+
+        if i == n-1: #last row
+
+            coeffs[i,i] = 1 #bn
+            coeffs[i, i-1] = 0 #an
+            funcs[i] = satellite[j] #fn
+
+        elif i != 0 and i != n-1:
+
+            D = [(T_list[j-1][i+1] - T_list[j-1][i-1])/2*delz \
+                 for i in range(n-1)]
+
+            rj = t/(4*delz) * (w[i] + 0.0057 * D[i] * kappa_firn[i] \
+            - (rho_func[i]*((kappa_firn[i]/rho[i]) + kappa_func[i])))
+
+            sj = (kappa_firn[i] * t)/(2*(delz**2))
+
+            coeffs[i,i] = 1+(2*sj) #bj
+            coeffs[i,i-1] = -rj - sj #aj
+            coeffs[i,i+1] = rj - sj #cj
+
+            funcs[i] = (sj - rj)*T_list[j-1][i+1] + \
+            (1-(2*sj))*T_list[j-1][i] + (sj + \
+            rj)*T_list[j-1][i-1] + \
+            (L[i] * (Lam*g/(c[i]*(rho[i]*1000)**3)) * (rho_func[i]) )#fj
+
+
+    T_list[j] = np.dot(np.linalg.inv(coeffs), funcs) #inverts the matrix
+    temps[j] = T_list[j][:] - 273.15 #converts all to Celsius, for plotting
+
+
+profile = temps[-4] #use profile corresponding to the month in which the ice core was drilled - May in this case
+#as satellite data extends a few months after core is drilled
+
+
+##PLOTTING
+
+recap = np.loadtxt(r'C:\Users\elloi\Documents\Uni\Bachelor\Projekt\recap.txt', delimiter=',') #loading recap data
+
+plt.figure()
+plt.plot(profile, z, label='Satellite data model')
+plt.plot(recap[:,1], H-recap[:,0]+48, label='RECAP data')
+#add 48m to recap data, as the H defined above subtracts 48m for dead ice
+plt.xlabel('Temperature (C)')
+plt.ylabel('Height above bedrock (m)')
+plt.legend()
+plt.show()