The Life Cycles of Some Nonlinear
Baroclinic Waves
Adrian J. Simmons and Brian J. Hoskins
Conor McNicholas, Ashly Spevacek, Jonathan Weyn
Outline
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Background
○ Historical
○ Motivation
Model
○ Framework
○ Initial conditions
Results
○ Synoptic features
○ Energetics
○ Eddy heat and momentum fluxes; meridional circulation
○ Zonal mean changes
Barotropic and quasi-barotropic considerations
Conclusions
Background
● 1940s: Seminal works of Charney (1947)
and Eady (1949)
● 1950s: First numerical simulations
(Charney et al. 1950; Phillips, 1956)
● 1970s: Expansion of linear baroclinic
instability with more complex numerical
simulations
○ Simmons and Hoskins (1976) and Gall
(1976a)
Jule Charney: Professor
of Meteorology, MIT
1956-1981
Motivation
Drawbacks of linear baroclinic instability
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Structure of linear solutions: weak
upper-level amplitudes (especially
for momentum fluxes).
○ Discrepancies of linear theory
concern in early climate
modelling (Schneider and
Dickinson, 1974; White 1977) .
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Previous studies by Simmons and
Hoskins: performed nonlinear
numerical integrations at low
resolution.
Model framework
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Global spectral model with T42
truncation (approx. 2.8 deg resolution) from Hoskins and Simmons (1975).
○ Vertical coordinate: sigma
coordinates
○ Vertical resolution: 14 layers
stacked closest to the surface
Assumptions: inviscid, adiabatic,
hydrostatic.
○ Internal diffusion & internal
smoothing.
○ Decay rate of (¼ day)-1
Initial conditions
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ICs consist of zonal, baroclinically unstable jets perturbed by small
amplitude disturbance of pre-determined normal mode.
○ Each mode: scaled to give initial sfc pressure wave of 1mb amplitude.
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Four flows examined, all at 200mb.
○ Two broad jets at 30ºN and 55ºN; two narrow jets at 45ºN and 30ºN.
○ Flows NOT intended to model specific observed/climatological profile
○ Four zonal wave #s for each initial disturbance
■ Wavenumber 6 perturbation for all jets
■ Wavenumber 9 perturbation for all jets excluding 55N broad jet
■ Wavenumber 12 and wavenumber 3 investigated for broad 30N
jet.
Synoptic Features
Pressure and temperature gradient
Wavenumber 6 perturbation effects on the 45 degree jet
● Surface low deepens 12 degrees north of the jet
maximum.
● A weak high pressure system develops south of the jet.
● Strongest temperature gradients located in positions
typical of occluded and cold fronts, with regions of
relatively warm air diminishing as the wave develops.
● Near the surface, the perturbation destroys the marked
temperature gradient between latitudes of 40 and 60
degrees and static stability is enhanced.
● At higher levels, little change occurs to the temperature
gradient.
Streamfunction and zonal velocity patterns
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The streamfunction develops a
southwest to northeast tilt to the
south of the disturbance
maximum.
By days 10 and 12 the easterlywesterly-easterly zonal wind
direction becomes very apparent.
The maximum strength of the
upper level zonal-mean flow is
almost constant throughout the
growth period.
Production of a stronger, narrower
jet develops as the wave decays.
Wavenumber 9 Differences
Wavenumber 9 perturbation effects on the 45 degree jet
● Surface effects are the same as those for the wavenumber 6 perturbation. Only
difference is variations in the amplitude.
● Shorter wavelength disturbance remains concentrated at low levels throughout the
course of the integration.
● At upper levels a relatively weak perturbation grows and decays retaining the
structure of the normal mode.
Energetics
Growth and decay
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In all cases growth of the disturbance is followed by a
period in which it decays by the same rate it grew.
Energy moves from zonal APE > eddy APE > eddy KE
Maximum energy values appear 3 days after the
temperature gradient first reverses at the surface.
Barotropic processes bring about a decay of amplitude.
Growth of the wave energy is limited largely by a local
stabilization of the flow, rather than by the overall
amount of available energy.
Larger energy values are associated with quick growth
rates.
Day 0
Key energetic features
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Internal dissipations cause the net loss of zonal
available potential energy to be slightly more than
twice the gain of zonal kinetic energy.
A substantially larger amplitude develops in the
upper atmosphere as opposed to close to the
surface. This is a result of a larger conversion of
eddy potential available potential energy to eddy
kinetic energy in the upper levels.
Day 10
Avg day 4 - 14
Eddy heat and
momentum fluxes
Eddy heat & momentum fluxes
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In general circulation statistics,
the upper-level amplitudes are
stronger than seen in linear
models
Wavenumber 6: heat flux
poleward
○ compared to linear model:
■ upper tropospheric
fluxes have much larger
amplitude
■ broader meridional
scale
Eddy heat & momentum fluxes
Eddy heat & momentum fluxes
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Wavenumber 6: momentum flux
poleward
○ predominantly in the upper
troposphere
Eddy heat & momentum fluxes
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Wavenumber 6: momentum flux
poleward
○ predominantly in the upper
troposphere
Eddy heat & momentum fluxes
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Wavenumber 9: heat flux poleward
○ predominantly in the lower troposphere
Momentum flux primarily equatorward, as
opposed to poleward for wavenumber 6
○ also much weaker amplitude
Eddy heat & momentum fluxes
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In the growth stage, the strength of the upper-level poleward momentum flux
increases more rapidly than that of the heat flux
In the decay stage, the heat flux vanishes 1+ days earlier than the momentum flux
○ momentum flux 2-3 times larger than for linear normal mode
Meridional circulation: enhanced upper-level poleward momentum flux enhances the
cell equatorward of the jet, reduces the polar cell
○ still acts to maintain thermal wind balance
Qualitatively, the similarity in structure between different zonal flows (jets) suggests
an improvement in using nonlinear theory
Nonlinear calculations enhance differences between wavenumbers 6 and 9
Zonal mean flow changes
Zonal mean flow
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The strength of the zonal jet increases
during the decay stage of the eddies
○ via rapid production of a smaller,
stronger jet
○ transfer of energy from eddy
momentum to zonal flow
○ this smaller jet is baroclinically
unstable
Qualitatively similar results for all jets
Wavenumber 9 disturbance has little
zonal mean change
Surface flow change sensitive to
friction
Zonal mean flow
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Reduction in temperature gradient
(especially at surface) during growth
stage
Temperature gradient increases
during decay to cancel out previous
change
○ consistent with thermal wind
balance of strengthening jet
QG considerations
Barotropic integrations
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Decay of disturbances is mainly barotropic
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I.C. for barotropic integration rom vort fields at ~300 mb (when eddy energy
is near max)
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Essential features of the decay are reproduced by the barotropic vorticity
equation.
○ Time scale of decay in barotropic solution is 1/2 that in the baroclinic
solution.
○ In full simulation, decay slowed by vertical motion acting to maintain
thermal wind balance in presence of strong vorticity advection.
Quasi-barotropic motion in baroclinic fluid
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Thermal wind + vorticity + temperature eq. in the QG system
yields:
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Growing mode: thermal and vorticity advection both influential
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Where eddy activity is maximized: low-level T-gradients
are weakened and the thermal forcing becomes weaker.
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T-advection plays a negligible role in forcing of vertical
motion during the decay stage.
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A 5-layer version of the QG model from Simmons and
Hoskins (1976) confirms this.
Day 10: Full QG Integration
Day 10: QG Integration;
Thermal Forcing Suppressed
Conclusions
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Nonlinearity significantly modifies aspects of linear theory
○ Linear theory can not be used to determine the spatial structure of eddy
fluxes.
■ Largest eddy fluxes occur close to time of maximum eddy energy
(when deviations from linear theory are greatest)
○ Large differences between short and long wavelength disturbances
pose a challenge to creating eddy-flux parameterization
■ Important task for climate modeling.
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Simmons and Hoskins (1978) stress: results should not be generalized
○ Baroclinic waves investigated in simple framework relative to the real
atmosphere.