Translating Susan's grid into NEMO parameters
Can we use the same number of vertical points, give up some resolution from 15 - 25 m to increase our resolution below to 250 m? Our current resolution:
level gdept_1d gdepw_1d e3t_1d e3w_1d
1 0.50 0.00 1.00 1.00
2 1.50 1.00 1.00 1.00
3 2.50 2.00 1.00 1.00
4 3.50 3.00 1.00 1.00
5 4.50 4.00 1.00 1.00
6 5.50 5.00 1.00 1.00
7 6.50 6.00 1.00 1.00
8 7.50 7.00 1.00 1.00
9 8.50 8.00 1.00 1.00
10 9.50 9.00 1.00 1.00
11 10.50 10.00 1.00 1.00
12 11.51 11.01 1.01 1.00
13 12.52 12.01 1.01 1.01
14 13.54 13.03 1.02 1.02
15 14.57 14.05 1.05 1.03
16 15.63 15.10 1.09 1.06
17 16.76 16.19 1.17 1.12
18 18.01 17.36 1.34 1.24
19 19.48 18.71 1.65 1.47
20 21.39 20.36 2.23 1.89
21 24.10 22.61 3.29 2.68
22 28.23 25.94 5.12 4.09
23 34.69 31.10 7.97 6.41
24 44.52 39.12 11.83 9.80
25 58.48 50.96 16.11 13.97
26 76.59 67.05 19.96 18.14
27 98.06 86.97 22.81 21.52
28 121.87 109.74 24.64 23.84
29 147.09 134.35 25.70 25.25
30 173.11 160.03 26.29 26.04
31 199.57 186.31 26.60 26.47
32 226.26 212.90 26.76 26.69
33 253.07 239.65 26.84 26.81
34 279.93 266.50 26.89 26.87
35 306.83 293.38 26.91 26.90
36 333.75 320.29 26.92 26.92
37 360.67 347.21 26.93 26.92
38 387.60 374.14 26.93 26.93
39 414.53 401.07 26.93 26.93
40 441.47 428.00 26.93 26.93
In [1]:
import matplotlib.pyplot as plt
import numpy as np
%matplotlib inline
In [2]:
# Original Grid Spacing
depth_orig = np.array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.01, 12.01, 13.03, 14.05, 15.1,
16.19, 17.36, 18.71, 20.36, 22.61, 25.94, 31.1, 39.12, 50.96, 67.05, 86.97, 109.74,
134.35, 160.03, 186.31, 212.9, 239.65, 266.5, 293.38, 320.29, 347.21, 374.14,
401.07, 428])
delta_depth_orig = np.zeros([39]); central_depth_orig = np.zeros([39])
delta_depth_orig = depth_orig[1:40] - depth_orig[0:39]
central_depth_orig = depth_orig[0:39] + delta_depth_orig/2.
In [3]:
def NEMO_tanh(ks, a, kth, acr):
"""A generic tanh used in NEMO"""
return a* np.tanh((ks-kth)/acr)
def NEMO_tanh_integral(ks, a, kth, acr):
"""A generic integrated tanh used in NEMO grid"""
return a*acr*np.log(np.cosh((ks-kth)/acr))
In [4]:
# Proposed Grid Spacing
# Top grid point, corrected for tails of tanh functions
a = 1-0.02068635-0.00153595-0.00000149177
# first tanh function (change across tanh, width, position)
b = 3; c = 6; d = 16.7 - 1.
# second tanh function
e = 13; f = 5.5; g = 22 - 1.
# third tanh function
h = 16; i = 5; j = 33.877- 1.
p = np.arange(39)
delta_depth_new = a + b/2.*(1+np.tanh((p-d)/c)) + e/2.*(1+np.tanh((p-g)/f)) + h/2.*(1+np.tanh((p-j)/i))
depth_new = np.zeros([40]); central_depth_new = np.zeros([30])
depth_new[0] = 0.
for ip in p:
depth_new[ip+1] = depth_new[ip] + delta_depth_new[ip]
central_depth_new = depth_new[0:39] + delta_depth_new/2.
print 'Surface cell Size = ',depth_new[1], 'Bottom Depth = ',depth_new[-1]
Surface cell Size = 0.999999999996 Bottom Depth = 428.000577161
Proposed grid spacing $$ z(k) = ppsur + ppa0k + ppa1 ppacr \log(\cosh(\frac{k-ppkth}{ppacr})) + ppa2 ppacr2 \log\cosh(\frac{k-ppkth2} {ppacr2})) + ppa3 ppacr3 \log\cosh(\frac{k-ppkth3} {ppacr3})) $$
$$ e3(k) = ppa0 + ppa1 \tanh(\frac{k-ppkth}{ppacr}) + ppa2\tanh(\frac{k-ppkth2}{ppacr2}) + ppa3\tanh(\frac{k-ppkth3}{ppacr3})$$Also need e3(1) = 1, z(1) = 0
In [5]:
ks = np.arange(1,41)
#translated from above
ppa1 = 3/2.; ppkth = 16.7; ppacr = 6.;
ppa2 = 13/2.; ppkth2 = 22.; ppacr2 = 5.5;
ppa3 = 16/2.; ppkth3 = 33.877; ppacr3 = 5.;
#setting e3(1) = 1
ppa0 = 1 - (NEMO_tanh(ks[0], ppa1, ppkth, ppacr) + NEMO_tanh(ks[0], ppa2, ppkth2, ppacr2) +
NEMO_tanh(ks[0], ppa3, ppkth3, ppacr3) )
#setting z(1) = 0
ppsur = -(NEMO_tanh_integral(ks[0], ppa1, ppkth, ppacr) + NEMO_tanh_integral(ks[0], ppa2, ppkth2, ppacr2) +
(NEMO_tanh_integral(ks[0], ppa3, ppkth3, ppacr3) + ppa0*ks[0]))
print 'ppa0', ppa0
print 'ppsur', ppsur
dz = ppa0 + NEMO_tanh(ks, ppa1, ppkth, ppacr) + NEMO_tanh(ks, ppa2, ppkth2, ppacr2) + NEMO_tanh(ks, ppa3, ppkth3, ppacr3)
z = ppsur + ppa0*ks + (NEMO_tanh_integral(ks, ppa1, ppkth, ppacr) + NEMO_tanh_integral(ks, ppa2, ppkth2, ppacr2) +
NEMO_tanh_integral(ks, ppa3, ppkth3, ppacr3))
print z[0], z[-1]
print dz[0], dz[-1]
ppa0 16.9777762082 ppsur -381.364772219 0.0 443.304590924 1.0 31.6859033266
In [6]:
fig, axs = plt.subplots(1,3,figsize=(15,10))
axs[0].set_ylim(top = depth_orig[0], bottom = depth_orig[-1])
axs[0].set_xlim(right = 33)
axs[1].set_ylim(top = depth_orig[0], bottom = 80.)
axs[1].set_xlim(right = 22)
axs[2].set_ylim(top = depth_orig[0], bottom = 30.)
axs[2].set_xlim(right = 7)
for ax in axs:
ax.plot(delta_depth_orig, central_depth_orig, label="Original")
ax.plot(delta_depth_new, central_depth_new, label="Proposed")
ax.plot(dz,z, label = 'Translated to NEMO')
ax.set_xlabel('Grid Spacing (m)')
ax.set_ylabel('Depth (m)')
ax.legend(); ax.grid()
In [ ]: