Stochastic Lattice Model - Preliminary Work

Model Overview
Below is our model, where H is mixed-layer heights (MASS), E liquid water static energy and Q moisture.

      dH/dt = physics(P) + advection(A) + interaction(I) + noise(N)
      dE/dt = physics(P) + advection(A) + interaction(I)
      dQ/dt = physics(P) + advection(A) + interaction(I)

  • Physics component is based on mixed-layer model physics. Entrainment is driven by both surface windsheer and buoyancy.  Cloud-top radiative cooling is coupled to LWP.  Drizzle is parameterized based on van Zanten et al (2005).
  • Advection component.  Right now, we have 4 choices, min mod, superbee, mc (monotonized centered), and van Leer. All of them seem to produce more diffusion than we expect.   This is seen in the following simple model runs. However noise and interaction components have much more priority than this one, so right now we just choose to leave it behind.
  • Interaction component.  We have 4 nearest-neighbor interaction based on mass and energy conservation law.  8-neighbor rule is expected to  be implemented.  The interaction is mainly promoted by drizzle process which induced local convergence.  The local enhancement of mass is balanced by nearest 4 neighboring sites, therefore energy and moisture are exchanged.
  • Noise component.  White Gaussian noise (WGN) is generated by Box-Muller method (Polar method, Ross, 1988) with zero mean and an amplitude comparable to the order of physics component.  Right now noise is only given to H field.


Preliminary Results
Simple model with no drizzle  (August,  2005)
In the figures, LWP is shown.  Dark color region are cloudy sites. White color region are cloud-free sites.  In this series of experiment, we just want to show that how different components of the model work. 

The interaction here has nothing to do drizzle.  It is a simple one that if more than the half of the neighbors are cloudy or clear, the local site will receive a tendency to change or keep its state.  The interaction and noise do not interfere much.
Apparently we do not want advection to be involved at this stage. 

 

Results with Drizzle Turned On (September, 2005)
 All the GIF animations below are the LWP fields.
  1. Illustration of  the interaction  rule 
  2. Randomly prescribed drizzle DND exp I
  3. Randomly prescribed drizzle DND exp II
  4. Prescribed drizzle DND with noise exp I
  5. Prescribed drizzle DND with noise exp II
  6. Uniform DND with noise exp I
  7. Uniform DND with noise exp II
  8. Randomly prescribed DND with noiseI
  9. Randomly prescribed DND with noiseII
Drizzle Droplet number densities (DND) for each experiment
1,4,5
 1000 in the background, 100 in the center region
2
 1000 in the background, 100-1000 random in the center
3,8,9
 1000 in the background, 100-300 random in the center region
6,7
 1000
Every experiment starts from uniform cloudy equilibrium states.
Noise level in 4,6,8 are comparable to physics component (entrainment rate), while in 5,7,9 it is 1/10 of the physics component.
 
Current thoughts
We turn on the parameterization of drizzle missing in the previous experiments.  After Exp1, I thought it may be the difference between the neighboring sites especially the corner regions of the two different DNDs that promotes the perturbation.  The interaction rule also helps the perturbation propagate and cluster.  So I continued with 2 & 3 to explore this.  It does so.  We do see some cloudy sites sorrouding by cloud-free sits and form diagonal oriented rows.  When I proceed to 4 & 5, I want to know what the role of the noise could be.  I found even in the uniform DND regions, the perturbation can grow and cluster.  It is not surprising since noise at each time step promotes the random difference.  But different noise levels give different spin-up times which is more obviously seen in 6 & 7.  When both noise and random DND are turned on, the role of noise does depends on the noise level.  This can be known by compare 8, 9 iwth 3.

Some concerns need to be discussed:
  • right now the interaction rule applies when negative physics components are present which we assume entrainment reduction because of drizzle.  However when noise is added, if negative, at next time step, it will also be enhanced to be positive as to simulate drizzle induced convergence.  so it is kind of mixed up....
  • when  after transition happens and the cloudy site is surrounding by cloud-free sites, it still draws mass from its neighbors  because of the interaction rule.   Thus the  cloud-free sites become shallower and shallower to some extent MLM can not handle.  So  we have to  figure out  when  the interaction should  be  stopped or reduced....


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