Spring, 2019
Jewett

This test is designed to help you evaluate your temperature advection code.
I recommend you start with test A2 -- Y-advection of theta.

Test A:  (Linear) advection

• Domain and execution:
• 33x33x16 grid - runs quickly for interactive testing
• (note odd dimension for nx,ny; bubble at exact X-Y center)
  
• ∆x=∆y=∆z = 500 meters
  
• ∆t = 1.0 seconds
  
• Physics:
• Advection of theta (only); piecewise linear (no monotonicity constraint on slopes)
• Strang splitting is now used in calling the x-, y-, and z- advection operators when advecting theta.
  
• Tests below were implemented by: setting U,V,W to constants (see below) every time step just after calls to my BC routine and before Advect.  Calls to diffusion and PGF/Buoyancy routines were commented out.
• I left calls to advection of U/V/W still in effect, but U,V,W were replaced every time step, so it did not matter.
  
• Initial condition:
• Theta: 1 thermal: -20˚, center (8250,8250,4250) meters (Fortran grid point 17,17,9); radius = 4000m in each direction x,y,z
• U,V,W: all are initially zero - no perturbations - other than changes mentioned below.
  
• Boundary conditions:
• BC routine is not changed (symmetric X, periodic Y, 0-gradient Z)
• For this to work, your usual BCs for U/V/W are overridden by changing your main program:
       >>>  set U, V, or W to a constant value every time step, just after BC is called.  Use the values listed below.
  

Test A1:  x-advection of theta

• Bubble placed at center at time T=0
• U = 20 m/s (constant); V=0, W=0
• This test requires "defeating" your U BCs by e.g. re-setting U2=20 everywhere after you call your BC routine ...
     ... to "override" where your BC routine flips the U values on the left and right boundary edges as it should.
  
• Bubble moves 8000m from X center [ 8250 m = ∆x/2 + (nx-1)/2•∆x ] to right domain edge [ 16250 m = ∆x/2 + (nx-1)•∆x ]
• At 20 meters per second, the bubble moves 8000m in 400 seconds.  Run code for 400s, plotting every 100s
• All plots made with plot3d program; theta contour plots below use contour interval of 1˚
  
• 3-D plots use plotp6 default "eye", looking toward origin from location in +X, +Y, +Z (so +X is out of page towards you)
  
• Text output from running code may be found here.  Animation of 3D view available as an animated GIF.
  

Test A1	T=0	T=100	T=200	T=300	T=400
X-Z j=17
X-Y k=9
3-D

Test A2:  y-advection of theta

• Bubble placed at center at time T=0; V = 20 m/s (constant); U=0, W=0
  
• Bubble moves 16000m from Y center [ 8250 m = ∆y/2 + (ny-1)/2•∆y ] to +Y domain edge [ 16250 m = ∆y/2 + (ny-1)•∆y ], out the periodic boundary at +Y top and re-emerging from periodic boundary at -Y bottom, returning to approximately the starting position (with slight lagging phase error)
  
• At 20 meters per second, the bubble moves 8000m from center to top Y edge in 400s, and from bottom periodic boundary edge to nearly the domain center in another 400s.   Run code for 800s, plotting every 200s
• All plots made with plotp6 program; theta contour plots below use contour interval of 1˚
  
• 3-D plots use plotp6 default "eye", looking toward origin from location in +X, +Y, +Z (so +X is out of page towards you)
  
• Text output from running code may be found here.  Animation of 3D view available as an animated GIF.

Test A2	T=0	T=200	T=400	T=600	T=800
Y-Z i=17
X-Y k=9
3-D

Test A3:  z-advection of theta

• Bubble placed at center at time T=0; w = -20 m/s (constant); U=0, V=0
  
• Bubble moves 4000m from near Z center [ 4250 m = ∆z/2 + nz/2•∆z ] to -Z bottom edge [ z = 250m = ∆z/2 ]
• This test requires "defeating" your W BCs by e.g. re-setting W=-20 everywhere after your call to your BC routine ...
     ... to "override" where your BC routine sets W values on the top and bottom boundaries to zero as it should.
• At 20 meters per second, the bubble moves 4000m from center to bottom in 200s.   Run code for 200s, plotting every 50s
• All plots made with plotp6 program; theta contour plots below use contour interval of 1˚
  
• 3-D plots use plotp6 default "eye", looking toward origin from location in +X, +Y, +Z (so +X is out of page towards you)
  
• Text output from running code may be found here.  Animation of 3D view available as an animated GIF.

Test A3	T=0	T=50	T=100	T=150	T=200
X-Z j=17
Y-Z i=17
3-D

Other advection tests

I don't currently have other advection tests on here.  There are several ways you could test advection of U, V, or W.  To test advection of U, for example (udu/dx + vdu/dy + wdu/dz), you could use the same formula you currently use for theta and use it to create a 'bubble' in U; then force V and/or W to be constant on every time step,  and see if it moves the U "bubble" correctly.  Another possibility is to set V=W=0, let U vary as sin(x), for example, and let it evolve; du/dt = -u*du/dx is a nonlinear problem that we investigated in computer program #1.

If you are having difficulty with your advection (of momentum, or theta), let me know - we can probably construct one of the above tests to check your code, if you can't find an obvious problem.