Vertical derivative diagnostics

We can automatically calculate the vertical derivative of a given field with respect to a vertical coordinate (pressure or height). In other words, the vertical derivative is the derivative of the field along a vertical coordinate. To automatically get the derivative of a field X, we can request X_pvert_derivative or X_zvert_derivative in the output yaml files.

WARNING: support for the dz-based derivative is experimental!

Brief description of algorithm

For simplicity, the field X can only have two dimensions (column and level). This covers most fields of interest, but notably does not cover higher-dimensioned fields in radiation and MAM. We calculate the derivative by finding the difference in the input field X, which must be provided on midpoints, by interpolating its values at interfaces weighted by the pressure differential. That is, for a pressure-based derivative

\[ \nabla_{p} X = \frac{\Delta X}{\Delta p} \]

and for height-based derivative

\[ \nabla_{p} X = \frac{\Delta X}{\Delta z} \]

where

\[ \Delta X = \tilde{X}_{i,k+1} - \tilde{X}_{i,k} \]

\[ \tilde{X}_{i,k+1} = \frac{\left( X_{i,k} \times \Delta p_{i,k+1} + X_{i,k+1} \times \Delta p_{i,k} \right)} {\left( \Delta p_{i,k+1} + \Delta p_{i,k} \right)} \]

Here, \(\Delta p\) is the "pseudo density" pressure differential in Pa; and \(\Delta z\) is the height of the layer in m

Note that the definition \(\tilde{X}_{i,k+1}\) assumes that all fields are weighted by the pressure differential, which is a prognostic parameter of the dynamics core solver. Also note that when \(\Delta p_{i,k+1} \gg \Delta p_{i,k}\), \(\tilde{X}_{i,k+1}\) is closer to \(X_{i,k}\) than \(X_{i,k+1}\).

Example

%YAML 1.1
---
filename_prefix: monthly.outputs
averaging_type: average
max_snapshots_per_file: 1
fields:
  physics_pg2:
    field_names:
      # in this example, we use T_mid in units of K
      - T_mid_pvert_derivative  # K / Pa
      - T_mid_zvert_derivative  # K / m
output_control:
  frequency: 1
  frequency_units: nmonths