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Avoid double staggering of ww in vertical advection of geopotential (w…
…rf-model#1338) TYPE: enhancement KEYWORDS: geopotential,vertical advection SOURCE: Matthias Göbel (University of Innsbruck) DESCRIPTION OF CHANGES: The geopotential equation in WRF is formulated in the following (advective) form where staggering/destaggering is marked with overbars: <img src="https://latex.codecogs.com/svg.latex?\partial_{t}\phi^{\prime}+\mu_{d}^{-1}[m_{x}m_{y}\left(\overline{\overline{U}^\eta\:\delta_{x}\phi}^x+\overline{\overline{V}^\eta\:\delta_{y}\phi}^y\right)+m_{y}\overline{\overline{\Omega}^\eta\:\delta_{\eta}\phi}^\eta-m_{y}gW]=0" title="http://latex.codecogs.com/svg.latex?\partial_{t}\phi^{\prime}+\mu_{d}^{-1}[m_{x}m_{y}\left(\overline{\overline{U}^\eta\:\delta_{x}\phi}^x+\overline{\overline{V}^\eta\:\delta_{y}\phi}^y\right)+m_{y}\overline{\overline{\Omega}^\eta\:\delta_{\eta}\phi}^\eta-m_{y}gW]=0" /> The vertical advection term thus employs a double staggering of Omega. First, Omega is destaggered to mass levels and multiplied with the phi gradient. Then the product is staggered back to w-levels. This formulation is analogous to the horizontal terms, but the double averaging of Omega may introduce additional inaccuracies and is not necessary. The phi gradient can be first staggered to w-levels and then multiplied with the unaveraged Omega: <img src="https://latex.codecogs.com/svg.latex?\overline{\overline{\Omega}^\eta\:\delta_{\eta}\phi}^\eta\rightarrow\Omega\:\overline{\delta_{\eta}\phi}^\eta" /> For consistency and to avoid numerical instabilities, the same thing has to be done in the acoustic step. The new namelist option `phi_adv_z` allows switching between the two formulations: `phi_adv_z = 1`: old formulation (default) `phi_adv_z = 2`: new formulation LIST OF MODIFIED FILES: Registry/Registry.EM_COMMON dyn_em/module_big_step_utilities_em.F dyn_em/module_small_step_em.F run/README.namelist TESTS CONDUCTED: Idealized test cases em_les and em_hill2d_x. The vertical velocity is affected by the modification. Updrafts and downdrafts in the em_hill2d_x case become ~10% stronger with the new formulation:  In the LES case stronger updrafts and downdrafts occur more frequently:  I came across this issue when trying to transform advective components of temperature, humidity, and momentum into Cartesian form in a consistent way, using w for the vertical advection. Without the proposed modifications I could not close the budget, because the vertical advection of phi was not consistent with the vertical advection in the other budget equations. RELEASE NOTE: A new namelist option is included that allows a user to avoiding double stagger averaging of Omega in the vertical advection of geopotential. The new namelist option phi_adv_z allows switching between the two formulations: phi_adv_z = 1 (the old formulation, which remains the default), and phi_adv_z = 2 (the new formulation).
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