ATSC 507 Homework 1   -   Hybrid-Eta Exercise 
R. Stull, Jan 2020.
Version 2

Use any computing method you want.   This problem can be done on a spreadsheet (with difficulty).  Or it might be easier to write your program in any of MatLab, R, python, or fortran, etc.

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Given:
2-D Domain:  xkm = 0 to 1000 km ,   zkm = 0 to 30 km
for your calculations, use dx = 20 km, and dz = 1 km or finer (I use dz=0.001 km).

Use the WRF definition of eta (eqs. 2.2 - 2.5 in WRF-ARW4 tech note 2019), where pi is dry hydrostatic pressure (written as pd in the 2019 tech note).
Let:  
pi_top = 2 kPa
eta_c = 0.3  . This is the eta value, above which eta becomes a pure pressure coordinate in this hybrid system.

For any location x(km), vertical profiles of temperature are given by
T(degC) =   (40 - 0.08*xkm)  -  6.5*zkm     for 0 < zkm < 12
T(degC) =   (40 - 0.08*xkm)  -  6.5*12       for 12 < zkm   (i.e., isothermal above 12 km)
for zkm = height above sea level

Actual Mean Sea-Level (at z = 0) pressure is Pmsl = 95kPa + 0.01*xkm   
Use the info above to determine P vs z, by iterating up from sea-level using hypsometric eq. 
P2 = P1 * exp[ (z1-z2) / (a*Tkelvin) ]   where a = 0.0293 km/K.  

This gives you the hydrostatic pressure (pi = P = pd) as a function of height.

NOTE: in WRF4 eq. (2.2), they use a REFERENCE sea-level pressure of Po = 100 kPa.  Yes, you should use this reference value in eq. (2.2), even though you use the actual Pmsl pressure as a basis for all your other calculations.

Intersecting the background pressure field that you just determined is the topography, 
given in height above sera level by:

Zground_km = 1 km + (1km)*cos[ 2*(3.14159)*(xkm-500km) / 500km ]     for  250km < x < 750 km
and Zground = 0 elsewhere.


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1) Find:  On an x-z graph, plot the altitudes (km) of the following isobaric surfaces:
100, 90, 80, 70, 60, 50, 40, 30, 20, 10, 5, 2 kPa.
On the same plot, plot the altitude of Zground.  


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2) Interpolate to find the Psurface (kPa) pressure at Zground. Namely, it is the pressure that corresponds to eta = 1.  
This pressure that you use to find eta in exercises (3) & (4).

Present the results in a table, where:  
row1 = x, (km)
row2 = Zground, (km)
row3=Psfc (kPa)


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3) Create a new P-x graph, on which you plot lines of constant eta, for the eta values listed below.  Namely, it should look something like WRF4 figure 2.1b, but with the more realistic meteorology that I prescribed above.  Also, like that figure, plot pressure P on the vertical axis in reversed order (highest pressure at the bottom of the figure), but don't use a log scale for P.

CAUTION: when calculating the values of B to use in WRF4 eq. (2.2), be advised that WRF4 eq. (2.3) applies only for eta > eta_c.    Otherwise, set B = 0 for eta <= eta_c.

eta = 
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.85
0.9
0.95
1


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4) Create a new z-x graph, on which you plot the z altitudes of the constant eta lines for the same eta values as in part (3) above. Make use of the hypsometric eq to find the heights z at the pressure levels that correspond to the requested eta values.

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