Potential Vorticity, Potential Temperature, Eqv. Latitude

Task

task	description
pv	potential vorticity
pvth	potential vorticity, potential temperature ('theta'), eqv. lat of pv on theta levels

Content versions

The content version is an attribute in hdf files.

content version	description
10	initial number pv (march 2003)
10	initial number pvth (oct 2003)

Fields

file	param	description	unit	shape	time res. (hr)	grib code	hor. comb.	vert. comb	Comments
pv	pv	potential vorticity	K m2/kg/s	im x jm x lm	[21,03], ..	060	mass aver	mass aver
pvth	pv	potential vorticity	K m2/kg/s	im x jm x lm	[21,03], ..		mass aver	mass aver
	theta	potential temperature	K	im x jm x lm	[21,03], ..		mass aver	mass aver
	eqvlatb	boundaries of eqv. lat. bins	deg	0:jm x ntheta	[21,03], ..
	eqvinds	index of eqv. lat. bin	index	im x jm x ntheta

Potential vorticity fields are valid for 6 hour intervals.

Computed from ECMWF fields valid for the mid of the interval. First computed on ECMWF's Gaussian Grid, then mass weighted average over cells and layers.

About potential vorticity

Definition

Definition according to [Holton, 1992, §4.3]:

$$P =  \left(\zeta_{\theta}+f\right)\left(-g\frac{\partial\theta}{\partial p}\right)$$ (4.1)

where

• $\theta$: potential temperature.
  Definition on page 52:
  

  $$\theta = T \left(\frac{p_0}{p}\right)^{\kappa}$$ (4.2)

  
  
  with
  • $p_0$ standard pressure (usually $10^5$ Pa)
  • $\kappa=R/c_p=2/7$
• $\zeta_{\theta}$: vertical component of relative vorticity evaluated on an isentropic surface $\theta=$constant.
  Definition on page 111:
  

  $$\zeta_{\theta} =  \vec{\mathbf{k}}\cdot\left(\vec{\mathbf{\nabla}}_{\theta}\times\vec{\mathbf{v}}\right)$$ (4.3)

  
  
  with $\vec{\mathbf{\nabla}}_{\theta}$ the gradient on an isentropic surface.
• $f=2\Omega\sin\phi$ is the Coriolis parameter:
  • $\Omega=7.292\cdot 10^-5 \mbox{rad/s}$: Earth's angular speed of rotation
  • $\phi$: latitude (rad)

Implementation

Gradients on isentropic surface.

Use equation 1.22 in [Holton, 1992] for horizontal gradient along isentropic surface:

$$\left(\frac{\partial F}{\partial x}\right)_{\theta} = \left(\frac{\partial F}{\partial x}\right)_{\eta}  +  \frac{\partial F}{\partial\eta}\left(\frac{\partial \eta}{\partial x}\right)_{\theta}$$ (4.4)

$$= \left(\frac{\partial F}{\partial x}\right)_{\eta}  -  \frac{\part... ...c{\partial\theta}{\partial x}\right)_{\eta} \frac{\partial\eta}{\partial\theta}$$ (4.5)

where first formula on page 22 is used for:

$$\left(\frac{\partial\eta}{\partial x}\right)_{\theta} = - \left(\frac{\partial\eta}{\partial\theta}\right)_{x} \left(\fra... ...c{\partial\theta}{\partial x}\right)_{\eta} \frac{\partial\eta}{\partial\theta}$$ (4.6)

Thus:

$$\zeta_{\theta} = \left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)_{\theta}$$ (4.7)

$$= \left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}... ...l \theta}{\partial y}\right)_{\eta} \right] \frac{\partial\eta}{\partial\theta}$$ (4.8)

Horizontal gradient of potential temperature:

$$\frac{\partial \theta}{\partial x} = \frac{\partial}{\partial x}\left[T\left(\frac{p_0}{p}\right)^{\kappa}\right]$$ (4.9)

$$= \frac{\partial T}{\partial x}\left(\frac{p_0}{p}\right)^{\kappa} ... ...0}{p}\right)^{\kappa} \frac{p}{p_0}\frac{p_0}{p^2}\frac{\partial p}{\partial x}$$ (4.10)

$$= \left(\frac{p_0}{p}\right)^{\kappa} \left[ \frac{\partial T}{\partial x} - \kappa T \frac{\partial\ln p}{\partial x} \right]$$ (4.11)

Pressure gradient of potential temperature:

$$\frac{\partial\theta}{\partial p} = \frac{\partial}{\partial p}\left[T\left(\frac{p_0}{p}\right)^{\kappa}\right]$$ (4.12)

$$= \frac{\partial T}{\partial p}\left(\frac{p_0}{p}\right)^{\kappa}  -  \kappa T \left(\frac{p_0}{p}\right)^{\kappa}\frac{p}{p_0}\frac{p_0}{p^2}$$ (4.13)

$$= \left(\frac{p_0}{p}\right)^{\kappa} \left[ \frac{\partial T}{\partial p} - \kappa \frac{T}{p} \right]$$ (4.14)

Potential vorticity:

$$P = -g \left(\zeta_{\theta}+f\right) \frac{\partial\theta}{\partial p}$$ (4.15)

$$= -g \left(\zeta_{\eta}  -  \left[ \frac{\partial v}{\partial\eta}\... ...rac{\partial\eta}{\partial\theta}  + f\right) \frac{\partial\theta}{\partial p}$$ (4.16)

$$= -g \left(\left(\zeta_{\eta} + f\right) \frac{\partial\theta}{\par... ...t] \frac{\partial\eta}{\partial\theta} \frac{\partial\theta}{\partial p}\right)$$ (4.17)

$$= -g \left(\left(\zeta_{\eta}+f\right) \frac{\partial\theta}{\parti... ...al u}{\partial p}\left(\frac{\partial \theta}{\partial y}\right)_{\eta} \right)$$ (4.18)

$$= -g \left(\frac{p_0}{p}\right)^{\kappa} \left( \left(\zeta_{\eta}+... ...tial T}{\partial y} - \kappa T \frac{\partial\ln p}{\partial y} \right] \right)$$ (4.19)

$$= -g \left(\frac{p_0}{p}\right)^{\kappa} \frac{1}{p} \left( \left(\... ...rac{\partial\ln p}{\partial y} \right] \frac{\partial u}{\partial\ln p} \right)$$ (4.20)

In terms of $\ln p_s$ this becomes:

$$P =  -g \left(\frac{p_0}{p}\right)^{\kappa} \frac{1}{p} ... ...\partial y} \right] \frac{\partial u}{\partial\ln p} \right)$$ (4.21)

Transformation to spherical coordinates using (3.20) gives:

$$P =  -g \left(\frac{p_0}{p}\right)^{\kappa} \frac{1}{p} ... ...partial\mu} \right] \frac{\partial u}{\partial\ln p} \right)$$ (4.22)

In old preprocessing, an analog of (4.20) was implemented with computed vorticity:

$$P =  -g \left(\frac{p_0}{p}\right)^{\kappa} \frac{1}{p} ... ...{\partial y}\right]\frac{\partial u}{\partial\ln p} \right)$$ (4.23)

PV from ECMWF?

060, PV, Potential vorticity, K m2/kg/s

In ECMWF archive since 2000-07-18 ?

Seems to be only on presure levels and potential temperature levels.

Potential temperature

3D field computed from temperature and pressure.

Definition according to [Holton, 1992, page 52]:

$$\theta = T \left(\frac{p_0}{p}\right)^{\kappa}$$ (4.24)

with

• $p_0$ standard pressure (usually $10^5$ Pa)
• $\kappa=R/c_p=2/7$

Equivalent latitude

Equivalent latitude of pv on some selected theta levels:

300.0, 315.0, 330.0, 350.0, 370.0, 395.0, 475.0, 600.0, 850.0 K