Vorticity Dynamics Including Diabatic Influence

Department of Meteorology

Potential vorticity dynamics including diabatic influence

Suzanne Gray

. A measure of the rotation in a fluid (the fluid analogue of angular velocity for a solid body) . Absolute vorticity = planetary vorticity + relative vorticity 휁 = 푓 + 휉 where the planetary vorticity is given by the Coriolis parameter: 푓 = 2Ω sin 휙 . In Cartesian (푥, 푦) coordinates: 휕푣 휕푢 휉 = − 휕푥 휕푦 where 푢, 푣 are the horizontal wind components

Recall potential temperature: 푅/푐푝 푝푟푒푓 휃 = 푇 푝 2 Potential vorticity conservation

. Parcel must remain confined between (bounded by) these two θ surfaces (adiabatic motion). . Mass is conserved: 훿푚 = 휌훿퐴훿ℎ = constant. 훿ℎ . Circulation is conserved: 훇 ∙ 퐧δA = constant where 훇 is absolute vorticity. 훿퐴 훇∙퐧 훇∙퐧 . Thus = constant and so 훿휃 = 휌훿ℎ 휌훿ℎ constant . Parcel of dry air moving between two nearby surfaces of constant θ (away from the ground so . In the limit of a infinitesimally small cylinder where friction is small) conserving its mass and 훿휃 훻휃 훻휃 = . Also 퐧 = . circulation around it. 훿ℎ |훻휃| . This yields the conservation of PV equation 퐷푃푉 1 = 0 where 푃푉 = 훇 ∙ 훻휃 퐷푡 휌

3 From Hoskins and James 2014 Potential vorticity conservation

PV is not conserved in the presence of frictional or diabatic heating effects. From the vorticity equation including these effects 퐷 1 1 ퟏ 1 훇 = 훇 ∙ 훁 퐮 − 훁흆 × 훁풑 + 훻 × 퐮ሶ 퐷푡 휌 휌 흆ퟑ 휌 it can be shown that……. 퐷 1 퐷푃푉 1 1 훇 ∙ 훁휽 ≡ = 훇 ∙ 훁휃ሶ + (훻 × 퐮ሶ ) ∙ 훻휃 퐷푡 휌 퐷푡 휌 휌

Diabatic Frictional processes processes

See Hoskins and James (2014) Chapter 10 for more details. 4 Potential vorticity invertibility

If the distribution of PV is known then the distribution of θ and wind can be deduced if 1. complete and consistent boundary conditions are specified around the edge of the flow domain (since a Poisson-like elliptical partial differential equation must be solved); 2. a balance condition linking the vorticity and θ fields is specified (e.g. geostrophic, gradient wind or Charney non-linear balance); 3. the total mass distribution is a known function of θ.

This is ‘PV thinking’, an extension of ‘vorticity thinking’. For the barotropic vorticity equation the streamfunction, 휓, is related to the relative vorticity, 휉, by the Poisson equation 훻2휓 = 휉 which implies 휓 ∝ −휉. Given the vorticity field at a given time this equation can be inverted to give the streamfunction. 5 PV inversion examples: I

1 푃푉 = 훇 ∙ 훻휃 = background 푃푉 + 푃푉 anomaly 휌

COLD WARM PV inversion examples: II

Cyclone X (anomalous θ PV is stippled)

X anticyclone

7 From Hoskins et al. (1985) PV inversion examples: III

θ Surface temperature W X anomalies act like PV anomalies

8 X C From Hoskins et al. (1985) Potential vorticity climatology 훇 ∙ 훻θ 푃푉 = 휌 tropopause (푓 + 훏) ∙ 훻θ = 휌 (푓 + ξ ) 휕휃 ≈ 푧 휌 휕푧 330K Potential vorticity is conserved 270K following fluid parcels for adiabatic frictionless flow.

This makes it a good tracer for Climatology of PV (in PVU, solid) and θ (dashed) in N. upper-tropospheric air over Hemi winter (Hoskins, 1990) several days. Planetary Rossby waves

. Wavelength on the order of thousands of kms

. Wave number is typically 3 to 6 (smaller wave number implies longer wavelength)

. Mountain ranges tend to disturb the upper-level wind flow; troughs are located east of such barriers

Some waves have small amplitude and flow is mostly zonal (W -> E). Others have larger amplitude and flow is mostly north to south. Planetary Rossby waves

. Large amplitude waves are always present in the atmosphere. . Dispersive waves propagating westwards relative to the mean flow. . They conserve potential vorticity . Exhibit stirring, stretching and folding properties including vortex roll up.

ECMWF PV on 315 K isentrope animation PV structure of dry cyclones +ve PV anomaly 1 . 푃푉 = 휁 ∙ 훻휃 conserved 휌 in frictionless adiabatic flow. . Role of PV in cyclone development through baroclinic instability. . Hoskins et al. (1985)

12 Heifetz et al. (2004) PV structure of dry cyclones

Idealised baroclinic waves

LC2: with added LC1 cyclonic shear

Potential temperature, Ɵ, on the PV = 2PVU surface (tropopause)

Thorncroft et al., (1993) PV structure of dry cyclones

. Modification of tropopause-level PV using water vapour imagery to improve short- range weather forecasts.

Satellite and PV

Original Modified

Demirtas and Thorpe (1999) 14 PV structure of dry cyclones

. Attribution of frictional generation of PV to Ekman and Baroclinic mechanisms. Frictional generation of PV through baroclinic mechanism in a growing wave

Positive baroclinic PV generation (low-level winds and thermal Adamson et al (2006) winds in opposing directions) Why diabatic processes?

The development of extratropical cyclones through (dry) baroclinic instability has been known about for more than 60 years (Eady, 1949, Charney, 1948).

Eric Eady (1915-1966) Jule Charney (1917-1981) BUT, cyclones are flows are moist!

17 Cyclone Ulli (also referred to as Cyclone Emil in Norway) in the North Sea off the eastern coast of Scotland on 3 January 2012 (NASA) Conceptual cyclones

Martínez-Alvarado et al. (2014) 18 Diabatic PV modification

퐷푔푃푉 휕 푔ℋ where H is the = 푓 2 heating rate 퐷푡 휕푧 휃0푁 Instantaneous heating Steady heating

PV dipole arising from heating applied in a barotropic environment Diabatic PV modification

퐷푔푃푉 휕 푔ℋ where H is the = 푓 2 heating rate 퐷푡 휕푧 휃0푁

CompositePV dipole midtropospheric arising fromPV heating (colours) applied and vertical in a velocity barotropic (contours)and forbaroclinic model simulated convectiveenvironment cells (rotated such that bulk wind shear is eastwards) with strong precipitation at the time Chagnonof maximum and ascent. Gray (2009) (Weijenborg et al. 2017) DRY MOIST PV structure of moist cyclones

Idealised baroclinic wave simulations.

PV on 316K (coloured); PV at 950 hPa (black contour for 1 PVU)

Schemm et al. (2013) 21 PV structure of moist cyclones

Direct effect: PV erosion and reduced upper-level wavelength. Indirect effect: jet enhancement leading to modified Rossby wave propagation

Plant et al. (2003) 22 PV structure of moist cyclones

. .Diagnosis of diabatically-generated PV, its attribution to modelled physical processes and influence on forecast evolution.

Reduced upper-level PV Diabatically-generated PV

Chagnon et al. (2013) Pomroy and Thorpe (2000) PV structure of moist cyclones

PV east-west vertical section PV anomaly composite (200 km through Lothar (1999) radius) NH DJF intense cyclones

24 Câmpa and Wernli (2012) PV influence on Rossby wave growth

Schematic illustration of the diabatic PV dipole relative to the tropopause in an evolving wave – based on extratropical cyclone case study

Chagnon et al. (2013) ∗ θ푒is the equivalent potential temperature that the air would Moist PV diagnostics have is saturated with water at the same T and p. θ푒 is conserved for reversible moist adiabatic processes. When air is ∗ 훇 ∙ 훻θ saturated θ푒 = θ푒 and hence 푃푉 = both are conserved. 휌 훇 ∙ 훻θ 푀푃푉 = 푒 휌 ∗ 훇푔 ∙ 훻θ푒 푀푃푉∗ = 푔 휌 ∗ where 훇푔is the geostrophic absolute vorticity andθ푒is the saturated equivalent potential temperature.

26 Atmospheric instability

27 From Schultz and Schumacher (1999) moist Mesoscale PV structure of cyclones

. Link between negative Slantwise convection at a front moist PV at fronts, Delta M -ve PVe conditional symmetric adjustment instability and frontal rainbands. ~zero PV Linear CSI

Nonlinear CSI

Holt and Thorpe (1991)

Browning et al. (2001)

Morcrette and Browning (2006) 28 Mesoscale PV structure: sting jets .Transient (few hours), mesoscale (~50km spread) jets of air descending from the tip of the hooked cloud head in the frontal fracture regions of some extratropical storms. .Can cause damaging winds (and especially gusts).

29 Adapted from Laura Baker by Neil Hart. Clark and Gray (2018). Summary

. “PV thinking” is a very useful approach to understanding atmospheric dynamics. . PV is conserved in frictionless adiabatic flow and the PV field can be inverted to yield the wind and 휃distribution given b.c.s and a balance constraint. . While dry dynamics can explain the growth of cyclones through baroclinic instability, lots of interesting things happen when moist processes are considered. . Diabatic processes in the cyclone warm conveyor belt can lead to I. a dipole of enhanced mid-tropospheric level PV with reduced PV below the tropopause and II. to a dipole of diabatic PV across the tropopause with positive diabatically- generated PV in the lower stratosphere (from LW cooling) and negative diabatic-generated PV in the upper troposphere . . Within the cyclone diabatic processes lead to I. enhanced intensification rates and intensity II. mesoscale instabilities associated with sting jets