CLUBB Verification Test

Component Being Tested

CLUBB (Cloud Layers Unified By Binormals), a parameterization of clouds and turbulence that is used in the E3SM v1 model for the representation of cloud macrophysics, shallow convection, and turbulence.

Purpose

Quantify the magnitude of clipping terms in CLUBB's 13 prognostic equations.

Description

The prognostic equations (i.e., time-evolution equations) that CLUBB solves in each grid cell contain a number of terms that do not correspond to any physical processes but are included for avoiding unphysical values or features in the discrete solution such as negative variances, correlation coefficients outside the range of [-1, 1], or artifitial extrema. For example, the equation describing the time evolution of $r_t'^{2}$, the subgrid variance of total water $r_t$ referred to as rtp2 in CLUBB's source code and model output, reads

\begin{equation}\begin{split} \\ \frac {\partial \overline{r_t^{\prime2}}} {\partial t} &= -\overline{w} \frac{\partial \overline{r_t^{\prime2}}}{\partial z} - \frac {1}{\rho_s} \frac {\partial \rho_s \overline{w^{\prime} r_t^{\prime2}}}{\partial z} - 2 \overline {w^{\prime}r_t^{\prime}} \frac{\partial \overline{r_t}}{\partial z} - \frac{C_2}{\tau} \Bigl( \overline{r_t^{\prime2}} - r_{t,tol}^{2} \Bigl ) \\ &\quad + \frac{\partial}{\partial z} \Bigr [ ( K_{w2} + \upsilon_2) \frac{\partial}{\partial z} \overline{r_t^{\prime2}} \Bigr ] + \overline{\frac {\partial {r_t^{\prime2}}} {\partial t} \Bigr |_{mc}} + \frac {\partial \overline{r_t^{\prime 2}}} {\partial t} \Bigr |_{pd} + \frac {\partial \overline{r_t^{\prime2}}} {\partial t} \Bigr |_{cl} \end{split}\end{equation}

The terms on the right hand side are tendencies caused by the mean advection, turbulent advection, turbulent production, dissipation term 1, dissipation term 2, microphysics, respectively, and two clipping terms indicated with subscripts $pd$ and $cl$.

The current version of CLUBB includes various types of clipping term, and not all types apply to all prognostic variables:

  • explicit, non-conservative clipping (denoted by subscript $cl$),
  • positive-definite conservative hole-filling ($pd$),
  • monotonic flux limiter ($mfl$),
  • semi-implicit clipping ($sicl$),
  • sponge layer damping ($sdmp$),
  • clipping due to the turbulent advection term ($tacl$).

In this verifcation test, the CLUBB stand-alone model is run for 6 model hours using the BOMEX (shallow convection) test case. We define the relative magnitude of a clipping term $F_{cl}$ of as

\begin{equation} F_{cl,rel} = \frac{ \sqrt{\sum_{i=1}^{I} \sum_{k=1}^{K} w_{k,i} (F_{cl,k,i}^2)}} { \max_{j = 1}^{J} \Bigr (\sqrt{\sum_{i}^{I} \sum_{k}^{K} w_{k,i} (F_{j,k,i}^2)} \, \, \Bigr )}, \end{equation}

with the weights $w_{k,i}$ being \begin{equation} w_{k,i} = \frac{m_{k,i}}{\sum_{k=1}^{K} m_{k,i}}, \quad m_{k,i} = \rho_{k,i}\Delta Z_{k,i}, \end{equation}

where $m$ and $\Delta Z$ are the mass and geometical thickness of layer $k$ at time step $i$, and $\rho$ denotes air density. $K$ is the total number of model layers. $I$ is the number of time steps used in the calculation. $j$ is the index of the physical terms. In other words, $F_{cl,rel}$ is the vertically integrated root-mean square magnitude of the clipping term $F_{cl}$ normalized by the largest magnitude of the physical terms. In the results shown below, the vertical integral covers the entire depth of the model atmosphere. The window of time averaging is either 0-3 h or 3-6 h after model initialization, with the former representing the spin-up phase and the latter being the quasi-equilibrium phase of a simulation.

Discussion

As the first step to designing a verification test that provides a "pass" or "fail" result, the BOMEX case was run with a range of step sizes: 2 s, 12 s, 30 s, 60 s (default), 120 s, 300 s, and 600 s. The relative magnitudes of the clipping terms are shown in the figure below. With the default step size of 60 s, the magnitude of the clipping terms are:

In [1]:
from IPython.core.display import display, Image
display(Image(filename='bomex_rms_clip_60s.png'))

Our current idea is to choose a threshold for each clipping term based on the results shown in the figure. Test simulations will be conducted with the default 60 s step size. For each clipping term, if the magnitude in a test simulation turns out to be smaller than the respective threshold, a "pass" is assigned to that particular term. An overall "pass" is issued when all terms are found to be smaller the corresponding thresholds.

Results of Initial Evaluation

In [2]:
display(Image(filename='relative_clipping_rms_weighted_bomex.png'))

Figure 1: Relative magnitude of the clipping terms. Simulations were conducted for the BOMEX case. Markers represent the 0-3 hr time average, circle markers represent the 3-6 hr time average. The explicit clipping term is in blue, positive definite terms are in green, and monotonic flux limiter terms are in red. Values less than 10$^{−6}$ are considered insignificant and not plotted.