Richard B. Rood (Room 2525, SRB) AOSS 401 Geophysical Fluid Dynamics: Atmospheric Dynamics Prepared: 20131121 Quasi-geostrophic / Analysis / Weather Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Cell: 301-526-8572

Class News Ctools site (AOSS 401 001 F13) Second Examination on December 10, 2013 Homework Homework due November 26, 2013

Weather National Weather Service Weather Underground Model forecasts: Weather Underground NCAR Research Applications Program

Outline Analysis of equations of atmospheric motion scaled for large-scale middle latitude dynamics / Quasi-geostrophic formalism Review of large-scale formalism Long and short waves / Barotropic waves Examine ageostrophic wind Cyclone development Occlusion Baroclinic  Barotropic /// energy conversion Vertical motion

Vorticity Equation DIVERGENCE TILTING SOLENOIDAL or BAROCLINIC Changes in relative vorticity are caused by: Divergence Tilting Gradients in density on a pressure surface Advection

QG Theory: Assumptions Assume the horizontal wind is approximately geostrophic Scale the material derivative Neglect the vertical advection Horizontal advection due to geostrophic wind Assume the north-south variation of the coriolis parameter is constant Divergence in the continuity equation only due to ageostrophic wind Modify the thermodynamic equation Advection by the geostrophic wind Assume hydrostatic balance Vertical velocity acts on a mean static stability profile

The quasi-geostrophic (QG) equations momentum equation geostrophic wind continuity equation thermodynamic equation

Geopotential tendency equation Vorticity Advection Thickness Advection Linear partial differential equation for geopotential tendency. Given a geopotential distribution at an initial time, can compute geopotential distribution at a later time. The right hand side is like a forcing.

First major set of conclusions from the quasi-geostrophic system We see that Geostrophic advection of geostrophic vorticity causes waves to propagate The vertical difference in temperature (thickness) advection causes waves to amplify

Remember our scaled vorticity equation? We see that the QG vorticity equation is very similar to the scaled vorticity equation we developed before …with a few additional assumptions

Long and Short Waves In this discussion these are both “large-scale” meaning that rotation is important. Therefore, this is a “short” large-scale wave versus a “long” large-scale wave.

Barotropic Wave Dispersion Look at the barotropic wave equation

Consider a barotropic fluid Think here How do we get to this form of the equation with geopotential? Barotropic: ageostrophic wind is zero, vertical velocity is zero, horizontal divergence is zero

Assume a “wave like solution” (get used to this…) Dispersion relation. Relates frequency and wave number to flow. Must be true for waves.

Wind must be positive, from the west, for a wave. Stationary wave? U positive for wave to exist. Return to lecture where we considered eastward and westward flow over the mountain. What is the wavelength for a stationary wave?  This leads to “planetary waves.” Scale order of 10,000 km. 1 – 2 – 3 of these around the planet, longitudinally. Wind must be positive, from the west, for a wave.

Consider a more specific form of a wave solution

Assume that the geopotential takes the form of a wave Mean Wave Mean Ф Gradient in y Horizontal wavenumber

Remember the relationship between geostrophic wind and geopotential Plug in the wave solution for the geopotential height

Divide the geostrophic wind into mean and perturbation Divide wind into mean and perturbation, assume no mean north/south wind Perturbation only mean perturbation

Plug into the advection of relative vorticity

Plug into the advection of planetary vorticity

Compare advection of planetary and relative vorticity

Advection of vorticity ζ < 0; anticyclonic  Advection of ζ tries to propagate the wave this way  ٠ ΔΦ > 0 B Φ0 - ΔΦ L L Φ0 H  Advection of f tries to propagate the wave this way  ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic

Advection of vorticity ζ < 0; anticyclonic  Short waves  ٠ ΔΦ > 0 B Φ0 - ΔΦ L L Φ0 H Long waves  ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic

More about the ageostrophic wind Review ageostrophic wind and implications for vertical motion and cyclone development Use all that we know to describe development of a mid-latitude cyclone

A closer look at the ageostrophic wind Start with our momentum equation Just for kicks, take and see what happens

A closer look at the ageostrophic wind Now, by the right hand rule: and remember so we can write

A closer look at the ageostrophic wind We end up with …the ageostrophic wind! Knowing that the divergence of the ageostrophic wind leads to vertical motion, let’s explore the implications of this…

Where do we find acceleration? Curvature Acceleration Ageostrophic wind ΔΦ > 0 Φ0 - ΔΦ D C Φ0 y, north Φ0 + ΔΦ x, east D = Divergence and C = Convergence

Where do we find acceleration? Along-flow speed change

Where do we find acceleration? Along-flow speed change Acceleration Ageostrophic wind D C J Remember our example of a thermal circulation, where divergence and convergence at different levels are related. D = Divergence and C = Convergence

Where do we find acceleration? Along-flow speed change

Vorticity at upper and lower levels Continue to examine the divergence of the wind  Which is a proxy for the vertical velocity

One more application… Start with the identity Now, consider the divergence of the ageostrophic wind Use the identity, and we have

What is this? Formally, this is Which is how we derived the vorticity equation

What can we do with this? Plug in our QG assumptions Let’s think about the difference in divergence (of the ageostrophic wind) between two levels By the continuity equation, this means that mass is either increasing in the column (net convergence) or decreasing in the column (net divergence) This should tell us whether low pressure or high pressure is developing at the surface…

Column Net Convergence/Divergence Subtract vorticity equation at 1000 hPa from vorticity equation at 500 hPa Gives us the net divergence between 1000 and 500 hPa It can be shown that

Examine each term “Steering term” Low-level centers of vorticity propagate in the direction of the thermal wind (Along the gradient of thickness)

Examine each term “Development term” A bit complicated Remember, thermal wind is the vertical change in the geostrophic wind This term indicates the influence of a tilt with height of the location of the maximum (minimum) in vorticity Fundamentally: if the location of the maximum in vorticity shifts westward with height, the low will develop.

Development term Combine terms Definition of thermal wind Definition of geostrophic vorticity

Development term Remember the barotropic height tendency equation? How about the equation?

Development term Development term is the vertical change in barotropic advection of vorticity This is the same as the stretching term in the omega equation If the upper-level wave propagates faster than the surface wave, the system decays Otherwise, the system may develop…

Implications Surface low and high pressure systems (centers of maximum/minimum vorticity) propagate along lines of constant thickness If there is a vertical tilt westward with height of the vorticity, then a surface low pressure system can intensify (increase in low-level positive vorticity)

Look at Cyclones

Mid-latitude cyclone development

Mid-latitude cyclone development From: http://www.srh.weather.gov/jetstream/synoptic/cyclone.htm

Mid-latitude cyclone development From: http://www.srh.weather.gov/jetstream/synoptic/cyclone.htm

Mid-latitude cyclone development From: http://www.srh.weather.gov/jetstream/synoptic/cyclone.htm

Mid-latitude cyclone development From: http://www.srh.weather.gov/jetstream/synoptic/cyclone.htm

Mid-latitude cyclone development From: http://www.srh.weather.gov/jetstream/synoptic/cyclone.htm

In the classic cyclone model Occlusion describes the transition of a cyclone from baroclinic (west-ward tilt with height) to barotropic (“vertically stacked”) Once there is no more westward vertical tilt with height, no further development can occur

Baroclinic  Barotropic

Idealized Development of a Baroclinic Wave Start with a N-S temperature gradient and associated wind speed maximum in the upper troposphere Introduce a low-level perturbation Watch the conversion of PE to KE and the development of the wave…

Energetics of Midlatitude Cyclone Development The jet stream is commonly associated with strong temperature gradients in the middle/lower troposphere (thermal wind relationship) Mid-latitude cyclones develop along waves in the jet stream By thermal wind balance, this means that mid-latitude cyclones develop along strong horizontal temperature gradients (fronts) There is a link between the strength of the horizontal temperature gradient and cyclone development…

Idealized vertical cross section

Two important definitions barotropic density depends only on pressure. By the ideal gas equation, surfaces of constant pressure are surfaces of constant density are surfaces of constant temperature (idealized assumption). = (p) baroclinic density depends on pressure and temperature (as in the real world). = (p,T)

Barotropic/baroclinic atmosphere p + p T+T p + 2p T+2T T T+T T+2T Baroclinic: p p + p p + 2p ENERGY HERE THAT IS CONVERTED TO MOTION

Barotropic/baroclinic atmosphere p + p T+T p + 2p T+2T T T+T T+2T Baroclinic: p p + p p + 2p DIABATIC HEATING KEEPS BUILDING THIS UP

Barotropic/baroclinic atmosphere Energetics: Baroclinic = temperature contrast = density contrast = available potential energy Extratropical cyclones intensify through conversion of available potential energy to kinetic energy

Available Potential Energy Defined as the difference in potential energy after an adiabatic redistribution of mass COLD WARM

Available Potential Energy Defined as the difference in potential energy after an adiabatic redistribution of mass WARM COLD

Energetics in the atmosphere Diabatic heating (primarily radiation) maintains the equator to pole temperature contrast Strength of temperature contrast referred to as “baroclinicity” Cyclones at mid-latitudes reduce this temperature contrast—adjust baroclinic atmosphere toward barotropic

Energetics in the atmosphere Ability to convert potential energy to kinetic energy directly related to tilt with height (offset) of low/high pressure

Energetics in the atmosphere Diabatic heating (primarily radiation) maintains the equator to pole temperature contrast Strength of temperature contrast referred to as “baroclinicity” Cyclones at mid-latitudes reduce this temperature contrast—adjust baroclinic atmosphere toward barotropic We can quantify this, but first let’s go back to the ageostrophic wind…

Baroclinic  Barotropic Transition

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Vertical velocity: Omega equation Kinematic method Horizontal advection Diabatic method Omega equation What are the ways that we think about vertical motion? Diabatic, Horizontal Advection, Kinematic Method.

Characteristics of large-scale vertical velocity In all of the estimates for vertical velocity what is missing? The answer is _______________ The vertical velocity in this large-scale, mid-latitude description of dynamics is exactly what is needed to maintain what balances ____________ ?

QG-omega equation Combine all QG equations Link between vertical derivative of vorticity advection (divergence/stretching) and vertical motion

QG-omega equation Combine all QG equations Link between vertical derivative of vorticity advection (divergence/stretching) and vertical motion Link between temperature advection and vertical motion

QG-omega equation Combine all QG equations Link between vertical derivative of vorticity advection (divergence/stretching) and vertical motion Link between temperature advection and vertical motion Link between diabatic heating and vertical motion

QG-omega equation (simplified) Advection of absolute vorticity by the thermal wind

“Advection” by thermal wind? How to analyze this on a map? Thermal Wind is Perpendicular to Gradient of Thickness Look at contours of constant thickness

Vertical Motion on Weather Maps Laplacian of omega is proportional to -ω Omega can be analyzed as: Remember, from definition of omega and scale analysis Positive vorticity advection by the thermal wind indicates rising motion

Vertical Motion on Weather Maps Positive vorticity advection by the thermal wind indicates rising motion Descent Ascent + Lines of constant thickness

Omega Equation

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Take Away Messages Large-scale dynamics / quasi-geostrophic theory Hydrostatic balance Geostrophic balance Vertical motion is important, but above balances are always maintained Vertical motion is diagnosed / no time dependence (though it changes with time) Vertical motion is linked to divergence of horizontal wind  ageostrophic wind Vertical motion works against static stability, which is large. This is how baroclinicity is represented in the equations, despite baroclinic terms being scaled out We have looked at dynamics of large-scale waves in different ways - heuristically, theoretically and on maps. We see consistency and different insights. The mathematics proves a way to both quantify and explore the behavior of the dynamics.