<SYSTEM_TASK:>
Given the following text description, write Python code to implement the functionality described below step by step
<END_TASK>
<USER_TASK:>
Description:
 <a href="http
Step1:  Now we set parameters
This part you will need to change for the different exercises. 
Note that Landlab does not impose units, but it assumes that all units are consistent. We will assume that everything is given in meters (m) and years (yr).
Step2:  Make a grid.
This part you may want to change.
Step3:  Set some variables related to time.
Step4:  Here we make the initial grid of elevation.
Step5:  Intializing all of the process components that do the work.
Step6:  Now for the code loop. Note that you can rerun this code block many times, and as long as you don't rerun any of the code boxes above, it will take the already evolved landscape and evolve it even more.
Step7:  Plot the topography.
Step8:  Plot the slope and area data at each point (in log-log space).
<ASSISTANT_TASK:>
Python Code:
# below is to make plots show up in the notebook
%matplotlib inline
# Code Block 1
import numpy as np
from matplotlib import pyplot as plt
from landlab import HexModelGrid, RasterModelGrid, imshow_grid
from landlab.components import (
    DepressionFinderAndRouter,
    FlowAccumulator,
    LinearDiffuser,
    StreamPowerEroder,
)
Explanation: <a href="http://landlab.github.io"><img style="float: left" src="../../../landlab_header.png"></a>
Modeling Hillslopes and Channels with Landlab
The original version of this exercise was donated by Andy Wickert at the University of Minnesota. This notebook was created by Nicole Gasparini at Tulane University.
<hr>
<small>For tutorials on learning Landlab, click here: <a href="https://github.com/landlab/landlab/wiki/Tutorials">https://github.com/landlab/landlab/wiki/Tutorials</a></small>
<hr>
What is this notebook?
This notebook illustrates a landscape evolution model in which the landscape evolves according to the equation:
\begin{equation}
 \frac{\partial z}{\partial t} = -K_\text{sp} A^{m_{sp}} S^{n_{sp}} + K_\text{hs} \frac{\partial^2 z}{\partial x^2} + U
\end{equation}
Here, $K$ are coefficients on the fluvial ($sp$) and hillslope ($hs$) parts of the equation, and $m_{sp}$ and $n_{sp}$ are positive exponents, usually thought to have a ratio, $m_{sp}/n_{sp} \approx 0.5$. $A$ is drainage area and $S$ is the slope of steepest descent ($-\frac{dz}{dx}$) where $x$ is horizontal distance (positive in the downslope direction) and $z$ is elevation. (If slope is negative there is no fluvial erosion.) $U$ is an externally-applied uplift field.
The first term on the right hand side of the equation is the fluvial erosion term, which is also known as the stream power equation. The second term on the right hand side of the equation is elevation changes via linear diffusion, and linear diffusion is one way in which to describe hillslope sediment transport.
For more information on the fluvial erosion term, please see:
Whipple, K.X. and Tucker, G.E., 1999. Dynamics of the stream‐power river incision model: Implications for height limits of mountain ranges, landscape response timescales, and research needs. Journal of Geophysical Research: Solid Earth.
For more information on linear diffusion applied to hillslopes (and other fun hillslope models) see:
Roering, J.J., 2008. How well can hillslope evolution models “explain” topography? Simulating soil transport and production with high-resolution topographic data. Geological Society of America Bulletin.
The ideas behind what this notebook does are presented nicely in the two papers below. Neither of them is exactly the same as this notebook, but they discuss drainage density and transitions from hillslope to fluviall processes.
Tucker, G.E. and Bras, R.L., 1998. Hillslope processes, drainage density, and landscape morphology. Water Resources Research.
Perron, J.T., Kirchner, J.W. and Dietrich, W.E., 2009. Formation of evenly spaced ridges and valleys. Nature.
What will you do?
In this exercise you will modify the code to get a better understanding of how different processes and forces control landscape evolution, landscape form and drainage density (as interpreted from slope-area data). It is expected that you have already learned the basics about fluvial bedrock incision (and the stream power equation) and sediment transport through creep on hillslopes (and the diffusion equation). (See references above.)
Start by sequentially running each code block without changing anything. To run an individual code cell, put the cursor in the cell and type shift-enter, or got to the Cell pulldown menu at the top and choose Run Cells. At the end of the notebook you will see the questions that you need to answer by changing parts of the code and rerunning it. 
Remember that you can always go to the Kernel pulldown menu at the top and choose Restart & Clear Output or Restart & Run All if you change things and want to start afresh. If you just change one code block and rerun only that code block, only the parts of the code in that code block will be updated. (E.g. if you change parameters but don't reset the code blocks that initialize run time or topography, then these values will not be reset.)  
Questions to answer before starting this assignment.
If hillslope diffusivity ($K_{hs}$) is fixed, but fluvial erodibility ($K_{sp}$) increases, what do you think will happen to the total relief and drainage density of a landscape?
If fluvial erodibility ($K_{sp}$) is fixed but hillslope diffusivity ($K_{hs}$) increases, what do you think will happen to the total relief and drainage density of a landscape?
If parameters stay fixed ($K_{hs}$ and $K_{sp}$), but the uplift rate increases, what do you think will happen to the total relief and drainage density of a landscape?
Now on to the code...
First we have to import the parts of Python and Landlab that are needed to run this code. You should not have to change this first code block.
End of explanation
# Code Block 2
uplift_rate = 0.001  # [m/yr], initially set at 0.001
K_sp = 1.0e-5  # units vary depending on m_sp and n_sp, initially set at 1e-5
m_sp = 0.5  # exponent on drainage area in stream power equation, initially 0.5
n_sp = 1.0  # exponent on slope in stream power equation, initially 1.
K_hs = 0.05  # [m^2/yr], initially 0.05
Explanation: Now we set parameters
This part you will need to change for the different exercises. 
Note that Landlab does not impose units, but it assumes that all units are consistent. We will assume that everything is given in meters (m) and years (yr).
End of explanation
# Code Block 3
ncells_side = 150  # number of raster cells on each side, initially 150
dxy = 50  # side length of a raster model cell, or resolution [m], initially 50
# Below is a raster (square cells) grid, with equal width and height
mg = RasterModelGrid((ncells_side, ncells_side), dxy)
# Below is a Hexagonal grid (hexagonal cells,
# with equal width and height (a square)
# mg = HexModelGrid(ncells_side, ncells_side, dxy, shape='rect')
# Below is a hexagonal grid (hexagonal cells),
# in which the grid is also a hexagon
# mg = HexModelGrid(ncells_side, ncells_side/2, dxy, shape='hex')
Explanation: Make a grid.
This part you may want to change.
End of explanation
# Code Block 4
dt = 1000  # time step [yr], initially 5000
total_time = 0  # amount of time the landscape has evolved [yr]
tmax = 1e6  # time for the model loop to run [yr], initially 1e6
t = np.arange(0, tmax, dt)  # each of the time steps that the code will run
Explanation: Set some variables related to time.
End of explanation
# Code Block 5
np.random.seed(0)  # seed set to zero so our figures are reproducible
mg_noise = np.random.rand(mg.number_of_nodes) / 1000.0  # intial noise on elevation grid
# set up the elevation on the grid
zr = mg.add_zeros("topographic__elevation", at="node")
zr += mg_noise
Explanation: Here we make the initial grid of elevation.
End of explanation
# Code Block 6
frr = FlowAccumulator(mg)  # intializing flow routing
spr = StreamPowerEroder(
    mg, K_sp=K_sp, m_sp=m_sp, n_sp=n_sp, threshold_sp=0.0
)  # initializing stream power incision
dfn = LinearDiffuser(
    mg, linear_diffusivity=K_hs, deposit=False
)  # initializing linear diffusion
# df = DepressionFinderAndRouter(mg) # Initializing the pit finder
Explanation: Intializing all of the process components that do the work.
End of explanation
# Code Block 7
for ti in t:
    zr[mg.core_nodes] += uplift_rate * dt  # uplift the landscape
    dfn.run_one_step(dt)  # diffuse the landscape
    frr.run_one_step()  # route flow
    # df.map_depressions()
    spr.run_one_step(dt)  # fluvial incision
    total_time += dt  # update time keeper
    print(total_time)
Explanation: Now for the code loop. Note that you can rerun this code block many times, and as long as you don't rerun any of the code boxes above, it will take the already evolved landscape and evolve it even more.
End of explanation
# Code Block 8
plt.figure(1)
imshow_grid(
    mg, "topographic__elevation", grid_units=("m", "m"), var_name="Elevation (m)"
)
title_text = f"$K_{{sp}}$={K_sp}; $K_{{hs}}$={K_hs}; $time$={total_time}; $dx$={dxy}"
plt.title(title_text)
max_elev = np.max(zr)
suptitle_text = "Maximum elevation is " + str(max_elev)
plt.suptitle(suptitle_text)
print("Maximum elevation is ", np.max(zr))
Explanation: Plot the topography.
End of explanation
# Code Block 9
plt.figure(2)
indices = np.where(mg.status_at_node[mg.at_node["flow__receiver_node"]] == 0)
plt.loglog(
    mg.at_node["drainage_area"][indices],
    mg.at_node["topographic__steepest_slope"][indices],
    "b.",
)
plt.ylabel("Topographic slope")
plt.xlabel("Drainage area (m^2)")
plt.title(title_text)
Explanation: Plot the slope and area data at each point (in log-log space).
End of explanation
<END_TASK>