(Click for larger image)
During the past week I have worked on the 'speed-based' meandering models (i.e. those that use water flow speed at the banks, rather than curvature or force, etc..., to calculate the meandering rate), including a simple centripetal (or 'sigma') model, and 5 variants of the Johannesson-Parker 1989 model. I have corrected all of the problems I had with the 4/13/08 code, and have made additional enhancements to the stand-alone Python river simulation program.
The simplest speed-based method approximates the channel at each point by a circular arc, and calculates the migration rate proportional to the excess speed at the outside bank. The ratio of bank speed to center (average) speed is (radius + width/2) / radius, the excess speed then being (in the JP notation):
u1b = ((radius + width/2) / radius - 1)
= (width/2) * (1/radius)
= b * C_tilde
= sigma
This method, combined with lag-averaging to yield an 'effective' downstream curvature, produces good results:
(Click for larger image)
The JP method modifies the basic sigma value to produce a more complicated expression for u1b which incldes terms from the part of the flow which is perpendicular to the stream direction. This method also includes a built-in lag and smoothing of sigma by virtue of being a second order differential (integral) system. The JP implementations I have tried are:
1. Integral equations from Lancaster 1998 and Lancaster and Bras 2002:
with
as in the Topographic Steering model.
2. Integral equations 45 and 46 from Johannesson-Parker 1989:
where A is the same as K in (1) above.
3. The same equations as in (2), but rather than performing the entire integrals for all values from 0 to phi, performing only one term of the integration per sigma_s and u_1b, and then resetting sigma_s_0 and u_1b_0 to the previous values from the last phi. This usually produced the same values for u_1b, but reduced the magnitudes of the integrands and integrals (due to the exponential terms) and thus stabilized the calculations for long streams (see below). It was also a bit faster to perform.
4. Instead of using the solutions in equations 45 and 46, solving the differential equations in 29c and 39:
Numerical solution is performed using the Runge-Kutta 4th order method on both equations separately. This was also somewhat faster to calculate than the integral solutions.
5. The equations in (4) above form a coupled second order system. Thus, they should be solved simultaneously, and as sigma and d(sigma)/d(phi) are allowed to vary during the time step. In this case, the solution is perfomed using a simultaneous interleaved RK4 method. This produced similar, but more stable, results (see below), at the expense of speed.
In general, implementations (1) - (5) form a second order initial value problem in either integral or differential equations.
To test these implementations, I created a circular channel with parameters of those in run F2 of Table 2 in JP89:
According to figure 5 of JP89, u1b for this configuration should be about 0.2 (i.e. ~1.2 times center speed). The circumferential speed method produces the following output (skip down to more pictures):
MeanderSC2: mode=3 mrate=0.000000 mexp=1.000000 ldist=0.000000 sdist=0.000000
i s curvature curvature_e speed rspeed espeed acc
----- -------------- -------------- -------------- -------------- -------------- -------------- --------------
0 0.000000 0.222222 0.222222 0.454545 1.111111 0.050505 0.045914
1 0.565115 0.222222 0.222222 0.454545 1.111111 0.050505 0.045914
2 1.130229 0.222222 0.222222 0.454545 1.111111 0.050505 0.045914
3 1.695344 0.222222 0.222222 0.454545 1.111111 0.050505 0.045914
4 2.260459 0.222222 0.222222 0.454545 1.111111 0.050505 0.045914
5 2.825573 0.222222 0.222222 0.454545 1.111111 0.050505 0.045914
6 3.390688 0.222222 0.222222 0.454545 1.111111 0.050505 0.045914
7 3.955803 0.222222 0.222222 0.454545 1.111111 0.050505 0.045914
8 4.520917 0.222222 0.222222 0.454545 1.111111 0.050505 0.045914
9 5.086032 0.222222 0.222222 0.454545 1.111111 0.050505 0.045914
10 5.651147 0.222222 0.222222 0.454545 1.111111 0.050505 0.045914
...
40 22.604587 0.222222 0.222222 0.454545 1.111111 0.050505 0.045914
41 23.169702 0.222222 0.222222 0.454545 1.111111 0.050505 0.045914
42 23.734816 0.222222 0.222222 0.454545 1.111111 0.050505 0.045914
43 24.299931 0.222222 0.222222 0.454545 1.111111 0.050505 0.045914
44 24.865046 0.222222 0.222222 0.454545 1.111111 0.050505 0.045914
45 25.430160 0.222222 0.222222 0.454545 1.111111 0.050505 0.045914
46 25.995275 0.222222 0.222222 0.454545 1.111111 0.050505 0.045914
47 26.560390 0.222222 0.222222 0.454545 1.111111 0.050505 0.045914
48 27.125504 0.222222 0.222222 0.454545 1.111111 0.050505 0.045914
49 27.690619 0.222222 0.222222 0.454545 1.111111 0.050505 0.045914
yielding a constant u1b of 0.111... (1/9). Pretty close, but a bit low. JP implementation (1) produces:
MeanderJP1: mode=0 mrate=0.000000 mexp=1.000000 ldist=0.000000 sdist=0.000000
i s curvature U C_f chi_20 A_s K bedslope integrand integral u_b1
----- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- --------------
0 0.000000 0.222222 0.454545 0.005218 1.056920 3.546911 0.000000 0.000000 2.222222e-01 0.000000e+00 0.106760
1 0.565115 0.222222 0.454545 0.005218 1.056920 3.546911 0.000000 0.000000 2.473727e-01 1.326875e-01 0.132797
2 1.130229 0.222222 0.454545 0.005218 1.056920 3.546911 0.000000 0.000000 2.753696e-01 2.803921e-01 0.156187
3 1.695344 0.222222 0.454545 0.005218 1.056920 3.546911 0.000000 0.000000 3.065351e-01 4.448135e-01 0.177200
4 2.260459 0.222222 0.454545 0.005218 1.056920 3.546911 0.000000 0.000000 3.412278e-01 6.278437e-01 0.196076
5 2.825573 0.222222 0.454545 0.005218 1.056920 3.546911 0.000000 0.000000 3.798469e-01 8.315886e-01 0.213032
6 3.390688 0.222222 0.454545 0.005218 1.056920 3.546911 0.000000 0.000000 4.228368e-01 1.058393e+00 0.228265
7 3.955803 0.222222 0.454545 0.005218 1.056920 3.546911 0.000000 0.000000 4.706922e-01 1.310866e+00 0.241949
8 4.520917 0.222222 0.454545 0.005218 1.056920 3.546911 0.000000 0.000000 5.239637e-01 1.591913e+00 0.254242
9 5.086032 0.222222 0.454545 0.005218 1.056920 3.546911 0.000000 0.000000 5.832644e-01 1.904769e+00 0.265285
10 5.651147 0.222222 0.454545 0.005218 1.056920 3.546911 0.000000 0.000000 6.492765e-01 2.253032e+00 0.275205
...
40 22.604587 0.222222 0.454545 0.005218 1.056920 3.546911 0.000000 0.000000 1.619408e+01 8.426355e+01 0.359343
41 23.169702 0.222222 0.454545 0.005218 1.056920 3.546911 0.000000 0.000000 1.802688e+01 9.393293e+01 0.359701
42 23.734816 0.222222 0.454545 0.005218 1.056920 3.546911 0.000000 0.000000 2.006710e+01 1.046967e+02 0.360022
43 24.299931 0.222222 0.454545 0.005218 1.056920 3.546911 0.000000 0.000000 2.233824e+01 1.166786e+02 0.360310
44 24.865046 0.222222 0.454545 0.005218 1.056920 3.546911 0.000000 0.000000 2.486641e+01 1.300166e+02 0.360569
45 25.430160 0.222222 0.454545 0.005218 1.056920 3.546911 0.000000 0.000000 2.768072e+01 1.448642e+02 0.360802
46 25.995275 0.222222 0.454545 0.005218 1.056920 3.546911 0.000000 0.000000 3.081354e+01 1.613922e+02 0.361011
47 26.560390 0.222222 0.454545 0.005218 1.056920 3.546911 0.000000 0.000000 3.430092e+01 1.797908e+02 0.361199
48 27.125504 0.222222 0.454545 0.005218 1.056920 3.546911 0.000000 0.000000 3.818300e+01 2.002716e+02 0.361367
49 27.690619 0.222222 0.454545 0.005218 1.056920 3.546911 0.000000 0.000000 4.250443e+01 2.230704e+02 0.361519
yielding a limiting value of about 0.36, which is a bit high. JP implementations (2) and (3) produce identical results:
MeanderJP2: mode=0 mrate=0.000000 mexp=1.000000 ldist=0.000000 sdist=0.000000
i s_tilde curvature U C_f F chi_20 A_s A bedslope sigma sigma_s integrand1 integral1 integrand2 integral2 u_1b
----- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- --------------
0 0.000000 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.111111 1.111111e-01 0.000000e+00 1.111111e-01 0.000000e+00 0.111111
1 0.565115 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.115088 1.236863e-01 6.634374e-02 1.281134e-01 6.759465e-02 0.116894
2 1.130229 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.116760 1.376848e-01 1.401961e-01 1.446853e-01 1.446759e-01 0.122588
3 1.695344 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117464 1.532675e-01 2.224068e-01 1.620304e-01 2.313407e-01 0.127912
4 2.260459 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117759 1.706139e-01 3.139218e-01 1.808226e-01 3.282163e-01 0.132784
5 2.825573 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117884 1.899234e-01 4.157943e-01 2.015002e-01 4.362444e-01 0.137197
6 3.390688 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117936 2.114184e-01 5.291964e-01 2.244049e-01 5.565870e-01 0.141177
7 3.955803 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117958 2.353461e-01 6.554330e-01 2.498489e-01 6.905909e-01 0.144759
8 4.520917 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117967 2.619819e-01 7.959567e-01 2.781479e-01 8.397803e-01 0.147980
9 5.086032 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117971 2.916322e-01 9.523844e-01 3.096380e-01 1.005863e+00 0.150874
10 5.651147 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117973 3.246382e-01 1.126516e+00 3.446867e-01 1.190748e+00 0.153474
...
40 22.604587 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117974 8.097040e+00 4.213177e+01 8.597168e+00 4.472877e+01 0.175534
41 23.169702 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117974 9.013438e+00 4.696647e+01 9.570170e+00 4.986208e+01 0.175628
42 23.734816 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117974 1.003355e+01 5.234833e+01 1.065329e+01 5.557637e+01 0.175712
43 24.299931 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117974 1.116912e+01 5.833930e+01 1.185900e+01 6.193739e+01 0.175787
44 24.865046 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117974 1.243321e+01 6.500831e+01 1.320117e+01 6.901832e+01 0.175855
45 25.430160 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117974 1.384036e+01 7.243210e+01 1.469523e+01 7.690065e+01 0.175916
46 25.995275 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117974 1.540677e+01 8.069609e+01 1.635840e+01 8.567508e+01 0.175971
47 26.560390 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117974 1.715046e+01 8.989538e+01 1.820979e+01 9.544258e+01 0.176020
48 27.125504 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117974 1.909150e+01 1.001358e+02 2.027072e+01 1.063155e+02 0.176065
49 27.690619 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117974 2.125222e+01 1.115352e+02 2.256490e+01 1.184190e+02 0.176104
MeanderJP3: mode=0 mrate=0.000000 mexp=1.000000 ldist=0.000000 sdist=0.000000
i s_tilde curvature U C_f F chi_20 A_s A bedslope sigma sigma_s integrand1 integral1 integrand2 integral2 u_1b
----- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- --------------
0 0.000000 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.111111 1.111111e-01 0.000000e+00 1.111111e-01 0.000000e+00 0.111111
1 0.565115 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.115088 1.236863e-01 6.634374e-02 1.281134e-01 6.759465e-02 0.116894
2 0.565115 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.116760 1.236863e-01 6.634374e-02 1.299751e-01 6.924440e-02 0.122588
3 0.565115 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117464 1.236863e-01 6.634374e-02 1.307579e-01 6.993815e-02 0.127912
4 0.565115 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117759 1.236863e-01 6.634374e-02 1.310872e-01 7.022988e-02 0.132784
5 0.565115 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117884 1.236863e-01 6.634374e-02 1.312256e-01 7.035256e-02 0.137197
6 0.565115 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117936 1.236863e-01 6.634374e-02 1.312838e-01 7.040415e-02 0.141177
7 0.565115 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117958 1.236863e-01 6.634374e-02 1.313083e-01 7.042584e-02 0.144759
8 0.565115 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117967 1.236863e-01 6.634374e-02 1.313186e-01 7.043497e-02 0.147980
9 0.565115 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117971 1.236863e-01 6.634374e-02 1.313229e-01 7.043880e-02 0.150874
10 0.565115 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117973 1.236863e-01 6.634374e-02 1.313247e-01 7.044041e-02 0.153474
...
40 0.565115 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117974 1.236863e-01 6.634374e-02 1.313260e-01 7.044159e-02 0.175534
41 0.565115 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117974 1.236863e-01 6.634374e-02 1.313260e-01 7.044159e-02 0.175628
42 0.565115 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117974 1.236863e-01 6.634374e-02 1.313260e-01 7.044159e-02 0.175712
43 0.565115 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117974 1.236863e-01 6.634374e-02 1.313260e-01 7.044159e-02 0.175787
44 0.565115 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117974 1.236863e-01 6.634374e-02 1.313260e-01 7.044159e-02 0.175855
45 0.565115 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117974 1.236863e-01 6.634374e-02 1.313260e-01 7.044159e-02 0.175916
46 0.565115 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117974 1.236863e-01 6.634374e-02 1.313260e-01 7.044159e-02 0.175971
47 0.565115 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117974 1.236863e-01 6.634374e-02 1.313260e-01 7.044159e-02 0.176020
48 0.565115 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117974 1.236863e-01 6.634374e-02 1.313260e-01 7.044159e-02 0.176065
49 0.565115 0.222222 0.454545 0.005218 0.619132 1.056920 3.546911 0.000000 0.000000 0.111111 0.117974 1.236863e-01 6.634374e-02 1.313260e-01 7.044159e-02 0.176104
yielding limiting values of about 0.18 --quite close. JP methods (4) and (5) also produce the same results:
MeanderJP4: mode=0 mrate=0.000000 mexp=1.000000 ldist=0.000000 sdist=0.000000
b H I U C_f r F
-------------- -------------- -------------- -------------- -------------- -------------- --------------
0.500000 0.055000 0.002000 0.454545 0.005218 10.541407 0.619132
chi_1 chi chi_20 delta A_s A
-------------- -------------- -------------- -------------- -------------- --------------
1.066004 0.732670 1.056920 16.159025 3.546911 0.000000
i x y phi curvature bedslope sigma sigma_s u_1b
----- -------------- -------------- -------------- -------------- -------------- -------------- -------------- --------------
0 4.500000 0.000000 0.000000 0.222222 0.000000 0.111111 0.111111 0.111111
1 4.464516 0.564000 0.565115 0.222222 0.000000 0.111111 0.111111 0.116489
2 4.358624 1.119104 1.130229 0.222222 0.000000 0.111111 0.111111 0.121319
3 4.183994 1.656560 1.695344 0.222222 0.000000 0.111111 0.111111 0.125659
4 3.943380 2.167892 2.260459 0.222222 0.000000 0.111111 0.111111 0.129557
5 3.640576 2.645034 2.825573 0.222222 0.000000 0.111111 0.111111 0.133060
6 3.280359 3.080462 3.390688 0.222222 0.000000 0.111111 0.111111 0.136206
7 2.868408 3.467310 3.955803 0.222222 0.000000 0.111111 0.111111 0.139032
8 2.411221 3.799476 4.520917 0.222222 0.000000 0.111111 0.111111 0.141571
9 1.916007 4.071722 5.086032 0.222222 0.000000 0.111111 0.111111 0.143851
10 1.390576 4.279754 5.651147 0.222222 0.000000 0.111111 0.111111 0.145900
...
40 1.390576 -4.279754 22.604587 0.222222 0.000000 0.111111 0.111111 0.163277
41 1.916007 -4.071722 23.169702 0.222222 0.000000 0.111111 0.111111 0.163351
42 2.411221 -3.799476 23.734816 0.222222 0.000000 0.111111 0.111111 0.163417
43 2.868408 -3.467310 24.299931 0.222222 0.000000 0.111111 0.111111 0.163477
44 3.280359 -3.080462 24.865046 0.222222 0.000000 0.111111 0.111111 0.163530
45 3.640576 -2.645034 25.430160 0.222222 0.000000 0.111111 0.111111 0.163578
46 3.943380 -2.167892 25.995275 0.222222 0.000000 0.111111 0.111111 0.163622
47 4.183994 -1.656560 26.560390 0.222222 0.000000 0.111111 0.111111 0.163660
48 4.358624 -1.119104 27.125504 0.222222 0.000000 0.111111 0.111111 0.163695
49 4.464516 -0.564000 27.690619 0.222222 0.000000 0.111111 0.111111 0.163726
MeanderJP5: mode=0 mrate=0.000000 mexp=1.000000 ldist=0.000000 sdist=0.000000
b H I U C_f r F
-------------- -------------- -------------- -------------- -------------- -------------- --------------
0.500000 0.055000 0.002000 0.454545 0.005218 10.541407 0.619132
chi_1 chi chi_20 delta A_s A
-------------- -------------- -------------- -------------- -------------- --------------
1.066004 0.732670 1.056920 16.159025 3.546911 0.000000
i x y phi curvature bedslope sigma sigma_s u_1b
----- -------------- -------------- -------------- -------------- -------------- -------------- -------------- --------------
0 4.500000 0.000000 0.000000 0.222222 0.000000 0.111111 0.111111 0.111111
1 4.464516 0.564000 0.565115 0.222222 0.000000 0.111111 0.111111 0.116489
2 4.358624 1.119104 1.130229 0.222222 0.000000 0.111111 0.111111 0.121319
3 4.183994 1.656560 1.695344 0.222222 0.000000 0.111111 0.111111 0.125659
4 3.943380 2.167892 2.260459 0.222222 0.000000 0.111111 0.111111 0.129557
5 3.640576 2.645034 2.825573 0.222222 0.000000 0.111111 0.111111 0.133060
6 3.280359 3.080462 3.390688 0.222222 0.000000 0.111111 0.111111 0.136206
7 2.868408 3.467310 3.955803 0.222222 0.000000 0.111111 0.111111 0.139032
8 2.411221 3.799476 4.520917 0.222222 0.000000 0.111111 0.111111 0.141571
9 1.916007 4.071722 5.086032 0.222222 0.000000 0.111111 0.111111 0.143851
10 1.390576 4.279754 5.651147 0.222222 0.000000 0.111111 0.111111 0.145900
...
40 1.390576 -4.279754 22.604587 0.222222 0.000000 0.111111 0.111111 0.163277
41 1.916007 -4.071722 23.169702 0.222222 0.000000 0.111111 0.111111 0.163351
42 2.411221 -3.799476 23.734816 0.222222 0.000000 0.111111 0.111111 0.163417
43 2.868408 -3.467310 24.299931 0.222222 0.000000 0.111111 0.111111 0.163477
44 3.280359 -3.080462 24.865046 0.222222 0.000000 0.111111 0.111111 0.163530
45 3.640576 -2.645034 25.430160 0.222222 0.000000 0.111111 0.111111 0.163578
46 3.943380 -2.167892 25.995275 0.222222 0.000000 0.111111 0.111111 0.163622
47 4.183994 -1.656560 26.560390 0.222222 0.000000 0.111111 0.111111 0.163660
48 4.358624 -1.119104 27.125504 0.222222 0.000000 0.111111 0.111111 0.163695
49 4.464516 -0.564000 27.690619 0.222222 0.000000 0.111111 0.111111 0.163726
both yielding values of about 0.16 --close but slightly low. Nevertheless, all implementations produce qualitatively similar results, and can be used as a basis for free-form meandering.
The differences bewteen implementations (1) - (2) and (3) becomes apparent for longer channels (in this case a river with 533 pts, length = 5638):
MeanderJP1: mode=0 mrate=15.000000 mexp=1.000000 ldist=15.000000 sdist=30.000000
i s curvature U C_f chi_20 A_s K bedslope integrand integral u_b1
----- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- --------------
0 0.000000 -0.009237 1.000000 0.024500 1.257704 22.430862 16.066791 0.371031 9.237218e-03 0.000000e+00 0.116177
1 8.515876 -0.009237 1.000000 0.024500 1.257704 22.430862 16.066791 0.371031 1.091515e-02 8.580754e-02 0.501640
2 17.030103 -0.008203 1.000000 0.024500 1.257704 22.430862 16.066791 0.329476 1.145297e-02 1.810312e-01 0.791399
3 25.693685 -0.008015 1.000000 0.024500 1.257704 22.430862 16.066791 0.321957 1.326288e-02 2.880951e-01 1.025024
4 34.673353 -0.010012 1.000000 0.024500 1.257704 22.430862 16.066791 0.402133 1.975360e-02 4.363336e-01 1.299782
5 49.681172 -0.009265 1.000000 0.024500 1.257704 22.430862 16.066791 0.372144 2.453221e-02 7.686503e-01 1.657444
6 57.588562 -0.011083 1.000000 0.024500 1.257704 22.430862 16.066791 0.445174 3.426605e-02 1.001121e+00 1.858204
7 65.437712 -0.012113 1.000000 0.024500 1.257704 22.430862 16.066791 0.486550 4.367927e-02 1.307023e+00 2.076378
8 73.414394 -0.010737 1.000000 0.024500 1.257704 22.430862 16.066791 0.431271 4.526855e-02 1.661777e+00 2.227235
9 81.345032 -0.008535 1.000000 0.024500 1.257704 22.430862 16.066791 0.342826 4.203654e-02 2.007970e+00 2.271458
10 89.917575 -0.008384 1.000000 0.024500 1.257704 22.430862 16.066791 0.336772 4.884959e-02 2.397532e+00 2.289769
...
515 5435.793672 -0.057902 1.000000 0.024500 1.257704 22.430862 16.066791 2.325728 1.079194e+45 3.436546e+46 10.515416
516 5445.535290 -0.054461 1.000000 0.024500 1.257704 22.430862 16.066791 2.187551 1.228630e+45 4.560643e+46 11.415946
517 5456.929557 -0.046431 1.000000 0.024500 1.257704 22.430862 16.066791 1.864992 1.309574e+45 6.006692e+46 11.888657
518 5468.158841 -0.043227 1.000000 0.024500 1.257704 22.430862 16.066791 1.736316 1.519384e+45 7.595050e+46 12.013807
519 5479.828125 -0.033457 1.000000 0.024500 1.257704 22.430862 16.066791 1.343885 1.478194e+45 9.344030e+46 11.647210
520 5493.621180 -0.028120 1.000000 0.024500 1.257704 22.430862 16.066791 1.129502 1.628041e+45 1.148625e+47 10.884833
521 5506.034701 -0.025226 1.000000 0.024500 1.257704 22.430862 16.066791 1.013251 1.862780e+45 1.365292e+47 10.131530
522 5518.348486 -0.023184 1.000000 0.024500 1.257704 22.430862 16.066791 0.931249 2.179351e+45 1.614162e+47 9.406704
523 5529.387537 -0.015922 1.000000 0.024500 1.257704 22.430862 16.066791 0.639521 1.858157e+45 1.837013e+47 8.555524
524 5537.622327 -0.014753 1.000000 0.024500 1.257704 22.430862 16.066791 0.592600 2.023418e+45 1.996833e+47 7.914040
MeanderJP2: mode=0 mrate=15.000000 mexp=1.000000 ldist=15.000000 sdist=30.000000
i s_tilde curvature U C_f F chi_20 A_s A bedslope sigma sigma_s integrand1 integral1 integrand2 integral2 u_1b
----- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- --------------
0 0.000000 -0.009237 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.371031 0.046186 0.046186 4.618609e-02 0.000000e+00 4.618609e-02 0.000000e+00 0.046186
1 8.515876 -0.009237 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.371031 0.046186 0.048143 5.457576e-02 4.290377e-01 5.688780e-02 4.388822e-01 0.175858
2 17.030103 -0.008203 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.329476 0.041013 0.046528 5.726485e-02 9.051559e-01 6.496414e-02 9.576198e-01 0.291809
3 25.693685 -0.008015 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.321957 0.040077 0.044542 6.631442e-02 1.440475e+00 7.370247e-02 1.558295e+00 0.381284
4 34.673353 -0.010012 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.402133 0.050058 0.048572 9.876801e-02 2.181668e+00 9.583670e-02 2.319497e+00 0.450383
5 49.681172 -0.009265 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.372144 0.046325 0.055256 1.226611e-01 3.843252e+00 1.463109e-01 4.136551e+00 0.592796
6 57.588562 -0.011083 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.445174 0.055415 0.055631 1.713303e-01 5.005604e+00 1.719974e-01 5.395045e+00 0.647231
7 65.437712 -0.012113 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.486550 0.060566 0.059602 2.183964e-01 6.535115e+00 2.149194e-01 6.913529e+00 0.703911
8 73.414394 -0.010737 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.431271 0.053685 0.059347 2.263428e-01 8.308886e+00 2.502141e-01 8.768640e+00 0.772123
9 81.345032 -0.008535 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.342826 0.042675 0.053230 2.101827e-01 1.003985e+01 2.621674e-01 1.080040e+01 0.826125
10 89.917575 -0.008384 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.336772 0.041922 0.048325 2.442479e-01 1.198766e+01 2.815589e-01 1.313095e+01 0.847045
...
515 5435.793672 -0.057902 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 2.325728 0.289508 0.201153 5.395972e+45 1.718273e+47 3.749186e+45 1.982556e+47 3.790496
516 5445.535290 -0.054461 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 2.187551 0.272307 0.267017 6.143150e+45 2.280322e+47 6.023796e+45 2.458579e+47 3.924331
517 5456.929557 -0.046431 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 1.864992 0.232155 0.269974 6.547869e+45 3.003346e+47 7.614533e+45 3.235573e+47 4.199535
518 5468.158841 -0.043227 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 1.736316 0.216137 0.253337 7.596918e+45 3.797525e+47 8.904423e+45 4.163053e+47 4.362581
519 5479.828125 -0.033457 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 1.343885 0.167287 0.215902 7.390972e+45 4.672015e+47 9.538841e+45 5.239152e+47 4.425828
520 5493.621180 -0.028120 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 1.129502 0.140601 0.181586 8.140206e+45 5.743127e+47 1.051309e+46 6.622038e+47 4.290710
521 5506.034701 -0.025226 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 1.013251 0.126130 0.156792 9.313901e+45 6.826461e+47 1.157813e+46 7.993187e+47 4.067128
522 5518.348486 -0.023184 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.931249 0.115922 0.140815 1.089676e+46 8.070810e+47 1.323666e+46 9.521007e+47 3.807346
523 5529.387537 -0.015922 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.639521 0.079608 0.111136 9.290784e+45 9.185066e+47 1.297037e+46 1.096751e+48 3.566168
524 5537.622327 -0.014753 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.592600 0.073767 0.092985 1.011709e+46 9.984165e+47 1.275287e+46 1.202664e+48 3.327349
MeanderJP3: mode=0 mrate=15.000000 mexp=1.000000 ldist=15.000000 sdist=30.000000
i s_tilde curvature U C_f F chi_20 A_s A bedslope sigma sigma_s integrand1 integral1 integrand2 integral2 u_1b
----- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- --------------
0 0.000000 -0.009237 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.371031 0.046186 0.046186 4.618609e-02 0.000000e+00 4.618609e-02 0.000000e+00 0.046186
1 8.515876 -0.009237 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.371031 0.046186 0.048143 5.457576e-02 4.290377e-01 5.688780e-02 4.388822e-01 0.175858
2 8.514227 -0.008203 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.329476 0.041013 0.046528 4.846180e-02 4.029268e-01 5.497752e-02 4.389945e-01 0.291809
3 8.663582 -0.008015 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.321957 0.040077 0.044542 4.749466e-02 3.833981e-01 5.278602e-02 4.302058e-01 0.381284
4 8.979667 -0.010012 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.402133 0.050058 0.048572 5.969078e-02 4.479422e-01 5.791923e-02 4.600354e-01 0.450383
5 15.007820 -0.009265 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.372144 0.046325 0.055256 6.216718e-02 8.421252e-01 7.415341e-02 9.209207e-01 0.592796
6 7.907390 -0.011083 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.445174 0.055415 0.055631 6.470523e-02 4.389782e-01 6.495718e-02 4.752875e-01 0.647231
7 7.849150 -0.012113 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.486550 0.060566 0.059602 7.063853e-02 4.947080e-01 6.951393e-02 4.911413e-01 0.703911
8 7.976683 -0.010737 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.431271 0.053685 0.059347 6.276963e-02 4.919043e-01 6.938965e-02 5.144615e-01 0.772123
9 7.930638 -0.008535 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.342826 0.042675 0.053230 4.985182e-02 4.105554e-01 6.218173e-02 4.818986e-01 0.826125
10 8.572543 -0.008384 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.336772 0.041922 0.048325 4.959159e-02 3.954799e-01 5.716713e-02 4.731918e-01 0.847045
...
515 9.546969 -0.057902 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 2.325728 0.289508 0.201153 3.490802e-01 1.845141e+00 2.425452e-01 1.753177e+00 3.790496
516 9.741617 -0.054461 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 2.187551 0.272307 0.267017 3.295955e-01 3.015532e+00 3.231919e-01 2.553985e+00 3.924331
517 11.394267 -0.046431 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 1.864992 0.232155 0.269974 2.902470e-01 3.204946e+00 3.375290e-01 3.444177e+00 4.199535
518 11.229284 -0.043227 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 1.736316 0.216137 0.253337 2.693490e-01 2.815765e+00 3.157066e-01 3.288384e+00 4.362581
519 11.669284 -0.033457 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 1.343885 0.167287 0.215902 2.102780e-01 2.487980e+00 2.713863e-01 3.061570e+00 4.425828
520 13.793056 -0.028120 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 1.129502 0.140601 0.181586 1.842454e-01 2.424355e+00 2.379532e-01 3.130026e+00 4.290710
521 12.413521 -0.025226 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 1.013251 0.126130 0.156792 1.608733e-01 1.871178e+00 1.999820e-01 2.368302e+00 4.067128
522 12.313786 -0.023184 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.931249 0.115922 0.140815 1.475651e-01 1.685111e+00 1.792523e-01 2.068991e+00 3.807346
523 11.039051 -0.015922 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.639521 0.079608 0.111136 9.883749e-02 1.185372e+00 1.379818e-01 1.538824e+00 3.566168
524 8.234790 -0.014753 1.000000 0.024500 0.202031 1.257704 22.430862 16.066791 0.592600 0.073767 0.092985 8.668793e-02 6.847051e-01 1.092726e-01 9.075100e-01 3.327349
Note 3 things about these results:
Differences between implementations (4) and (5) become apparent when the curvature changes rapidly (in this case, a rapidly varying sine channel):
b H I U C_f r F
-------------- -------------- -------------- -------------- -------------- -------------- --------------
5.000000 2.500000 0.001000 1.000000 0.024500 102.040816 0.202031
chi_1 chi chi_20 delta A_s A
-------------- -------------- -------------- -------------- -------------- --------------
0.491935 0.158602 1.257704 11.077316 22.430862 16.066791
i x y phi curvature sigma sigma_s u_1b
----- -------------- -------------- -------------- -------------- -------------- -------------- --------------
0 0.000000 0.000000 0.000000 -0.000886 0.004431 0.004431 0.004431
1 10.101010 24.868989 26.842075 -0.000886 0.004431 0.004431 0.036630
2 20.202020 48.175367 52.243211 -0.002212 0.011060 0.004431 0.051601
3 30.303030 68.454711 74.898942 -0.004888 0.024442 0.007016 0.051642
4 40.404040 84.432793 93.802104 -0.011537 0.057686 0.018307 0.049375
5 50.505051 95.105652 108.497012 -0.029053 0.145265 0.047038 0.052341
6 60.606061 99.802673 119.636689 -0.054420 0.272101 0.114199 0.085023
7 70.707071 98.228725 129.859591 -0.043068 0.215341 0.218189 0.405344
8 80.808081 90.482705 142.588740 -0.018345 0.091723 0.216153 1.320990
9 90.909091 77.051324 159.394465 -0.007415 0.037073 0.127727 2.201127
10 101.010101 58.778525 180.273296 -0.003283 0.016414 0.078872 2.320520
...
90 909.090909 -58.778525 1773.804280 0.003283 0.016414 0.078806 1.749844
91 919.191919 -77.051324 1794.683111 0.007415 0.037073 0.045181 1.653657
92 929.292929 -90.482705 1811.488836 0.018345 0.091723 0.039419 1.388391
93 939.393939 -98.228725 1824.217985 0.043068 0.215341 0.076813 1.140559
94 949.494949 -99.802673 1834.440887 0.054420 0.272101 0.168044 1.080596
95 959.595960 -95.105652 1845.580564 0.029053 0.145265 0.239191 1.517369
96 969.696970 -84.432793 1860.275472 0.011537 0.057686 0.170662 2.389702
97 979.797980 -68.454711 1879.178633 0.004888 0.024442 0.097462 2.721875
98 989.898990 -48.175367 1901.834365 0.002212 0.011060 0.068988 2.438207
99 1000.000000 -24.868989 1927.235501 0.002212 0.011060 0.066606 2.007624
0 0.000000 0.000000 0.000000 -0.000886 0.004431 0.004431 0.004431
1 10.101010 24.868989 26.842075 -0.000886 0.004431 0.004431 0.037325
2 20.202020 48.175367 52.243211 -0.002212 0.011060 0.010961 0.054616
3 30.303030 68.454711 74.898942 -0.004888 0.024442 0.022260 0.108652
4 40.404040 84.432793 93.802104 -0.011537 0.057686 0.046421 0.211897
5 50.505051 95.105652 108.497012 -0.029053 0.145265 0.102164 0.384101
6 60.606061 99.802673 119.636689 -0.054420 0.272101 0.186757 0.681656
7 70.707071 98.228725 129.859591 -0.043068 0.215341 0.219885 1.317798
8 80.808081 90.482705 142.588740 -0.018345 0.091723 0.156976 1.979930
9 90.909091 77.051324 159.394465 -0.007415 0.037073 0.077242 2.052744
10 101.010101 58.778525 180.273296 -0.003283 0.016414 0.039847 1.640476
...
90 909.090909 -58.778525 1773.804280 0.003283 0.016414 0.038628 0.141001
91 919.191919 -77.051324 1794.683111 0.007415 0.037073 0.042403 0.215542
92 929.292929 -90.482705 1811.488836 0.018345 0.091723 0.071978 0.352442
93 939.393939 -98.228725 1824.217985 0.043068 0.215341 0.145755 0.562941
94 949.494949 -99.802673 1834.440887 0.054420 0.272101 0.214659 1.001298
95 959.595960 -95.105652 1845.580564 0.029053 0.145265 0.198811 1.733755
96 969.696970 -84.432793 1860.275472 0.011537 0.057686 0.112220 2.131258
97 979.797980 -68.454711 1879.178633 0.004888 0.024442 0.054139 1.900176
98 989.898990 -48.175367 1901.834365 0.002212 0.011060 0.031298 1.392894
99 1000.000000 -24.868989 1927.235501 0.002212 0.011060 0.030466 0.899074
Realistic examples of these differences will be shown below. Here is the code for all 5 JP meandering implementations. I apologize for the current paucity of comments:
MeanderJP1.py | MeanderJP2.py | MeanderJP3.py | MeanderJP4.py | MeanderJP5.py
I should also mention that setting the 'Use RK4' checkbox also has an effect on the result. Although not always the case, in general a river produced using the Runge-Kutta 4 integration method will be longer and have more detail (with the same migration rate and number of steps, started in the same initial configuration) than one not using this method, although it will take about 2x as long to produce. Note that RK4 integration can be applied to all meandering models (although both the JP4 and JP5 implementations also have another level of RK4 integration internally which is used to solve the differential equations). Here is an example of using the 'global' RK4 method (right):
(Click for larger image)
Here is a comparison of running the centripetal, JP1, JP3, and JP4 methods for 500 steps on the same initial configuration for width = 5.5, depth = 0.5, flow = 1.6, slope = 0.0014, and grain size = 0.0007 (all from Lancaster & Bras 2002):
(Click for larger image)
So, which one is right? 8^)
This completes the work for April, in which I have accomplished the following:
In May I will work on: