A Toy Lagrangian Thermal Model

In a recent paper of mine in JAS I describe a toy model of a convecting thermal in a moist atmosphere. It’s a fun little model to play with so I thought I’d share it here in simpler terms with some extra examples.

Hannah, W. M., 2017: Entrainment vs. Dilution in Tropical Deep Convection. J. Atmos. Sci.74, 3725–3747.

The idea for this model comes from an age old problem about entrainment, which is just a fancy term for the process of mixing relatively dry, non-cloud air into a convecting thermal (i.e. a cloud). We have a sense of how large the rate of entrainment should be, but for convective parameterizations in climate models we are forced to use much smaller values to get realistic behavior. In other words, there seems to be an “effective entrainment” that is much smaller than the actual entrainment.

My paper is one of several that suggest the air between the cloud and the completely cloud-free air plays an important role in this respect. Clouds are very turbulent and this causes air to be re-entrained many times (Yeo and Romps 2012), which effectively recycles the humid air. Furthermore, the cloud tends to form a shell of humid air at the edge of the cloud so that dry air being entrained becomes more humid by the time it reaches the core of the cloud (Dawe and Austin 2011).

Using these ideas I wanted to construct a toy cloud model based on the simplified cartoon above. I’m mainly interested in the movement of the core of the thermal, but I want to include a passive shell of air that is created from mixing the dry environmental air with the air expelled out of the core (i.e. detrainment). The core entrains directly from this mixture in the shell, instead of entraining directly from the environment.

Model Derivation

I’ll skip over the details of the derivation, but I think it can be helpful to see the equations. First of all, here are the equations of the model if there is only a thermal core and no shell.

These three equations predict the thermal’s mass, thermodynamics, and momentum, respectively. Here’s a quick legend of the symbols:

m  –  mass
Φ  –  energy tracer (specifically, moist static energy)
w  –  velocity
B  –  buoyancy
D –  drag
ε  –  entrainment
δ  –  detrianment

The subscript “c” indicates a property of the thermal core.

To add a shell to this thermal we only need to address the first two equations above, because we will make the simplifying assumption that the shell is non-buoyant and does not have an appreciable affect on the momentum of the thermal core. So by adding an equation for the shell mass and energy we get the final model equations shown below.

The subscript “s” indicates a property of the shell. Notice in the 2nd and 4th equations we have separate entrainment and detrainment rates of the shell and core. Currently, we don’t have a good way to specify those rates, so for now we’ll just prescribe fixed values.

When turning these model equations into code there’s a few things to watch out for.

  • assumed shape for drag calculation
  • numerical discretization and timestep
  • mass going to zero or negative – balance of mass source and sink
  • entrainment units – using a fixed length scale and converting to a mixing timescale requires using the vertical velocity, however this can be tricky when the thermal reverses direction. In other words, the sign of entrainment should not switch when the sign of vertical velocity switches. This isn’t a caveat of the model, just a caveat of the wonky way we need to get around the fact that we don’t know how to diagnose entrainment.

Since the shell acts as a “buffer” to protect the core against the effects of entrainment, I call this model the Buffered Lagrangian Thermal model, or the BLT for short.

I’m not going to post the code here, but I’m happy to share the model code with anyone who is wants to play around with it. Just shoot me an email if you’re interested.

Model Results

I’ll just briefly talk about some interesting aspects of the model. The plot below shows the cloud height and vertical velocity for four simulations. The values for the entrainment and detrainment rates are fixed.

The blue line shows a thermal without a shell, whereas the red line shows a thermal with a shell. The difference in height and updraft speed is pretty large. The oscillation at the top is just due to the thermal bouncing around it’s level of neutral buoyancy. I actually added a bit of artificial damping for this plot just to damp this oscillation out quicker, but this is actually observed in real clouds. In real clouds, these oscillations probably don’t last long because detainment should become very strong at the level of neutral buoyancy. However, this simple model doesn’t exhibit that behavior.

So the effect of the shell is pretty dramatic, but you might be asking, “couldn’t you get the same result by lowering the entrainment rate?” The answer is… sort of. The dashed blue line shows a thermal with no shell, but lower entrainment. The cloud reaches a similar height as the red line, but velocity and thermodynamic characteristics of the thermal are very different. You might say that they each take a different “thermodynamic path” to get to the same height.

Another interesting aspect of this model is that when the entrainment and detrainment rates are set to different values we can get some interesting sensitivities. The dashed green curve shows an example where the core detrainment is enhanced relative to the core entrainment. The resulting thermal goes almost twice as high! The reason is because the stronger detrainment moistens the shell and further decreases the dilution by entrainment, resulting in a more buoyant thermal.

There a lot of simplifications used in this model, but I’m hoping to build on this idea a bit more and maybe someday use this type of cloud model in a convective parameterization. The most difficult part of doing this is determining how many thermals to “launch” at a time. The size and thermodynamic properties of each thermal also needs to be determined. This is related to the long-standing “closure problem” faced by other convective parameterizations. My hope is that using a different kind of cloud model would inspire a new approach to solving the closure problem.


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