RID diagnostics explained

The RID provides various data divided into three components:

Component	Label
Zonal-mean data on pressure levels	zonal
Horizontal slices of selected pressure levels	single-level
Surface variables *under development	surface

Zonal

The zonal-mean component of the dataset is composed of the following subcomponents:

Label	Description
core	Core set of zonal-mean variables
fluxes	Eddy covariance terms based on zonal anomalies
dt	Temporal derivatives
mom	Diagnostics based on the momentum equation
tem-qg	Transformed Eulerian mean diagnostics based on quasi-geostrophic equations
tem-pr	Transformed Eulerian mean diagnostics based on primitive equations
tem-thermo	Thermodynamic diagnostics based on Transformed Eulerian mean primitive equations
moist	Diagnostics involving specific humidity

Notation: Overbars ($\overline{x}$) denote zonal averages and primes ($’$) departures therefrom.

Zonal diagnostics are provided on the following pressure levels:

core: zonal mean of core variables

Symbol	Description	NetCDF name	Units
$\overline{u}$	Zonal-mean zonal wind	u	$m\,s^{-1}$
$\overline{v}$	Zonal-mean meridional wind	v	$m\,s^{-1}$
$\overline{\omega}$	Zonal-mean pressure velocity	w	$Pa\,s^{-1}$
$\overline{T}$	Temperature	t	$K$
$\overline{Z}$	Geopotential height	z	$m$

fluxes: eddy covariance terms based on zonal anomalies

Symbol	Description	NetCDF name	Units
$\overline{u’^2}$	Zonal wind variance	uu	$m^2\,s^{-2}$
$\overline{v’^2}$	Meridional wind variance	vv	$m^2\,s^{-2}$
EKE	Eddy kinetic energy	obtained by adding uu and vv	$m^2\,s^{-2}$
$\overline{Z’^2}$	Variance of geopotential height	zz	$m^2$
$\overline{T’^2}$	Temperature variance	tt	$K^2$
$\overline{u’v’}$	Covariance of zonal and meridional wind anomalies, or meridional mementum flux	vu	$m^2\,s^{-2}$
$\overline{u’\omega’}$	Covariance of zonal wind anomalies and pressure-velocity wind anomalies, or meridional momentum flux	wu	$m\,Pa\,s^{-2}$
$\overline{v’T’}$	Covariance of meridional wind anomalies and temperature anomalies, or meridional heat flux	vt	$m\,K\,s^{-1}$
$\overline{\omega’T’}$	Covariance of pressure velocity anomalies and temperature anomalies, or vertical heat flux	wt	$Pa\,K\,s^{-1}$

dt: temporal derivatives

Symbol	Description	NetCDF name	Units
$\partial{}\overline{T}/\partial{}t$	Time derivative of temperature	t_dt	$K\, s^{-1}$
$\partial{}\overline{u}/\partial{}t$	Time derivative of zonal wind	u_dt	$m\,s^{-2}$

mom: diagnostics of the momentum equation

Symbol	Description	NetCDF name	Units
$f\overline{v}$	Acceleration by Coriolis torque	coriolis_torque	$m\,s^{-2}$
$ – \overline{v}\frac{1}{a\cos\phi}\frac{\partial{}(\overline{u}\cos\phi)}{\partial\phi}$	Acceleration by meridional advection of momentum	u_adv_by_v	$m\,s^{-2}$
$- \overline{\omega}\frac{\partial\overline{u}}{\partial{}p} $	Acceleration by vertical advection of momentum	u_adv_by_w	$m\,s^{-2}$
$-\frac{\partial{}(\overline{\omega’u’})}{\partial{}p}$	Acceleration by vertical momentum flux convergence	u_accel_by_wu_flux	$m\,s^{-2}$
$- \frac{1}{a\cos^2\phi}\frac{\partial(\overline{u’v’}\cos^2\phi)}{\partial\phi}$	Acceleration by meridional momentum flux convergence	u_accel_by_vu_flux	$m\,s^{-2}$

In the zonal mean, the momentum equation is written:

$$\frac{\partial\overline{u}}{\partial{}t} = \overbrace{f\overline{v}}^{\text{coriolis_torque}} \underbrace{- \overline{v}\frac{1}{a\cos\phi}\frac{\partial{}(\overline{u}\cos\phi)}{\partial\phi}}_{\text{u_adv_by_v}} \overbrace{-\overline{\omega}\frac{\partial\overline{u}}{\partial{}p}}^{\text{u_adv_by_w}} \underbrace{- \frac{1}{a\cos^2\phi}\frac{\partial(\overline{u’v’}\cos^2\phi)}{\partial\phi}}_{\text{u_accel_by_vu_flux}}\overbrace{-\frac{\partial{}(\overline{\omega’u’})}{\partial{}p}}^{\text{u_accel_by_wu_flux}} $$

tem-qg and tem-pr: diagnostics of Eliassen-Palm flux and transformed Eulerian mean

Symbol	Description	NetCDF name	Units
$\phi$	latitude	lat
$p$	pressure	pre
$a$	Earth radius, 6371000 m		$m$
$\overline{v}^*$	Meridional component of residual circulation	vres	$m\,s^{-1}$
$\overline{\omega}^*$	Vertical component of residual circulation	wres	$Pa\,s^{-1}$
$f\overline{v}^*$	Coriolis torque	coriolis_torque_tem	$m\,s^{-2}$
$- \overline{v}^*\frac{1}{a\cos\phi}\frac{\partial{}(\overline{u}\cos\phi)}{\partial{}\phi}$	Momentum advection by meridional residual circulation	u_adv_by_vres$^{1}$	$m\,s^{-2}$
$- \overline{\omega}^*\frac{\partial\overline{u}}{\partial{}p}$	Momentum advection by vertical residual circulation	u_adv_by_wres$^{1}$	$m\,s^{-2}$
$F_p$	vertical component of EP-flux	EPF_pre_[qg,pr]	$Pa\,m^{2}\,s^{-2}$
$F_\phi$	meridional component of EP-flux	EPF_lat_[qg,pr]	$m^{3}\,s^{-2}$
$\frac{(\nabla\cdot\mathbf{F})_p}{a\cos\phi}$	EP-flux divergence impact on zonal wind acceleration, vertical component	EPFD_pre_[qg,pr]$^{2}$	$m\,s^{-2}$
$\frac{(\nabla\cdot\mathbf{F})_\phi}{a\cos\phi}$	EP-flux divergence impact on zonal wind acceleration, meridional component	EPFD_lat_[qg,pr]$^{2}$	$m\,s^{-2}$
$\frac{\nabla\cdot\mathbf{F}}{a\cos\phi}$	EP-flux divergence impact on zonal wind acceleration	obtained by adding EPFD_pre_[qg,pr] and EPFD_lat_[qg,pr]	$m\,s^{-2}$

In the transformed Eulerian mean, the momentum equation is written

$$\frac{\partial{}\overline{u}}{\partial{t}} = \overbrace{f\overline{v}^*}^{\text{coriolis_torque_tem}} \underbrace{- \overline{v}^*\frac{1}{a\cos\phi}\frac{\partial{}(\overline{u}\cos\phi)}{\partial{}\phi}}_{\text{u_adv_by_vres}} \overbrace{- \overline{\omega}^*\frac{\partial\overline{u}}{\partial{}p}}^{\text{u_adv_by_wres}} + \frac{\nabla\cdot\mathbf{F}}{a\cos\phi}$$

where the residual circulation is defined as

$$\overbrace{\overline{v}^*}^{\text{vres}}=\overline{v}-\frac{\partial}{\partial{}p}\left[\frac{\overline{v’\theta’}}{\partial{}\overline{\theta}/\partial{}p}\right]$$

$$\overbrace{\overline{\omega}^*}^{\text{wres}}= \overline{\omega}+\frac{1}{a\cos\phi}\frac{\partial}{\partial\phi}\left[\frac{\overline{v’\theta’}\cos\phi}{\partial\overline{\theta}/\partial{}p}\right]$$

The equation of the EP-flux divergence is:

$$\nabla\cdot\mathbf{F}=\underbrace{ \frac{1}{a\cos{}\phi} \frac{\partial{}(\overbrace{F_\phi}^{\text{EPF_lat_[qg,pr]}}\cos\phi)}{\partial{}\phi}}_{(\nabla\cdot\mathbf{F})_\phi}+\underbrace{\frac{\partial{}\overbrace{F_p}^{\text{EPF_pre_[qg,pr]}}}{\partial{}p}}_{(\nabla\cdot\mathbf{F})_p}$$

In the dataset, the EP-flux divergence is provided in the form of its impact on zonal wind acceleration. It is therefore scaled by $\frac{1}{a\cos\phi}$:

$$\overbrace{\frac{(\nabla\cdot\mathbf{F})_p}{a\cos\phi}}^{\text{EPFD_pre_[qg,pr]}}= \frac{1}{a\cos\phi}\frac{\partial{}\overbrace{F_p}^{\text{EPF_pre_[qg,pr]}}}{\partial{}p}$$

$$\overbrace{\frac{(\nabla\cdot\mathbf{F})_\phi}{a\cos\phi}}^{\text{EPFD_lat_[qg,pr]}}= \frac{1}{(a\cos{}\phi)^2} \frac{\partial{}(\overbrace{F_\phi}^{\text{EPF_lat_[qg,pr]}}\cos\phi)}{\partial{}\phi}$$

EP-flux is defined as follows (terms only included in the primitive-equation version are indicated with “pr”)

$$\{F_{\phi},\,F_p\} = a\cos\phi\left\{\overbrace{\frac{\overline{v’\theta’}}{\partial\overline{\theta}/\partial{}p}\frac{\partial\overline{u}}{\partial{p}}}^{\text{pr}}-\overline{u’v’} , \, -\overbrace{\frac{\overline{v’\theta’}}{\partial\overline{\theta}/\partial{}p}\frac{1}{a\cos\phi}\frac{\partial{}(\overline{u}\cos\phi)}{\partial\phi}}^{\text{pr}}+\frac{\overline{v’\theta’}}{\partial{}\overline{\theta}/\partial{}p}f-\overbrace{\overline{\omega’u’} }^{\text{pr}}\right\} $$

For a reference with these equations, although in a slightly different form, refer to Martineau, P., Wright, J. S., Zhu, N. and Fujiwara, M.: Zonal-mean data set of global atmospheric reanalyses on pressure levels, Earth Syst. Sci. Data, 10(4), 1925–1941, doi:10.5194/essd-10-1925-2018, 2018.

Wavenumber decompositions are provided for EP-flux in the NetCDF files in the following order: full field, wavenumber 1, wavenumber 2

tem-thermo: thermodynamic equation from the transformed Eulerian mean

Symbol	Description	NetCDF name	Units
$-\overline{v}^*\frac{1}{a}\frac{\partial\overline{\theta}}{\partial{}\phi}$	Advection of potential temperature by the meridional residual circulation	theta_adv_by_vres	$K \,s^{-1}$
$ – \overline{\omega}^*\frac{\partial\overline{\theta}}{\partial{}p}$	Advection of potential temperature by the vertical residual circulation	theta_adv_by_wres	$K \,s^{-1}$
$-\frac{\partial}{\partial{}p}\left(\frac{\overline{v’\theta’}\frac{\partial\overline{\theta}}{\partial\phi}}{a\frac{\partial\overline{\theta}}{\partial{}p}}+\overline{\omega’\theta’}\right)$	Potential temperature tendency due to eddy fluxes	flux_term	$K \,s^{-1}$

In the transformed Eulerian mean, the thermodynamic equation takes the following form:

$$\frac{\partial\overline{\theta}}{\partial{}t} = \overbrace{-\overline{v}^*\frac{1}{a}\frac{\partial\overline{\theta}}{\partial{}\phi}}^{\text{theta_adv_by_vres}} \underbrace{- \overline{\omega}^*\frac{\partial\overline{\theta}}{\partial{}p}}_{\text{theta_adv_by_wres}}\overbrace{-\frac{\partial}{\partial{}p}\left(\frac{\overline{v’\theta’}\frac{\partial\overline{\theta}}{\partial\phi}}{a\frac{\partial\overline{\theta}}{\partial{}p}}+\overline{\omega’\theta’}\right)}^{\text{flux_term}}+Q$$

moist: diagnostics involving specific humidity

Symbol	Description	NetCDF name	Units
$\overline{q}$	Specific humidity	q	$kg\, kg^{-1}$
$\overline{v’q’}$	Meridional flux of specific humidity / meridional moisture flux	vq	$m\,s^{-1}\,kg\,kg^{-1}$
$\overline{\omega’q’}$	Vertical flux of specific humidity / vertical moisture flux	wq	$Pa\,s^{-1}\,kg\,kg^{-1}$

Single-level

Variable	Description	Levels (hPa)
$T$	Temperature	850
$q$	Specific humidity	850
$Z$	Geopotential height	500, 10
$u$	Eastward wind / zonal wind	850

Surface

Under development