1. A Stochastic Model with a Low-frequency Amplification Feedback for the Stratospheric Northern Annular Mode
15 minutes
Nanjing university of information, Science, and technology
FLORIDA STATE UNIVERSITY
Yueyue Yu
WCRP/SPARC Workshop 2017
Incheon, South Korea

2. OVERVIEW Motivation Data and Indices
Mass budget analysis: linking the NAM with Meridional Mass Circulation Variability
A Linear Stochastic Model for the NAM Variability
Summary

3. See-saw pattern in P (z) and T fields
Motivation
Stratospheric NAM
Jet (u) variability
Quasi-geostrophic balance
TEM theory
Wave fluxes
See-saw pattern in P (z) and T fields
EOF1 of 50 hPa monthly height anomalies
Percentage Variance explained:46%
Negative phase ~ Cold in mid-latitudes
This is the pattern of NAM in the stratosphere, familiar
TEM has explained how Waves interact with jet via momentum budget analysis and other fields associated with NAM can be inferred from the wind side of the QG equation.
However, the dynamic and thermodynamic process in this direction is not directly revealed.
It would be beneficial to look at the whole story of the NAM variability from both directions separately.
The Mass circulation theory is a nice tool to do both mass and momentum budget and combine them without QG requirement.
In this study, we focus on understanding this part. And that part of study is on going.
Meridional Mass Circulation
(MMC) theory

4. Meridional Mass Circulation
Jan 1979
Poleward warm air branch
Townsend and Johnson (1985); Johnson (1989); Held and Schneider (1999)；Cai and Shin（2014); Cai et al. (2016)
Westward tilted waves ~
Potential Temp (K)
Equatorward cold air branch
Johnson （1989）
Please allow me to introduce mass circulation a little bit.
Generalized Lagrangian meridional circulation using potential temp as vertical coordinate
2 advantages:
This allows us to investigate the mass budget over a given region due to adiabatic and diabatic processes separately.
Reveal the relationship between NAM and stratospheric mass transport, both of which is related to cold events at surface. N S
Semi-Lagrangian view of meridional circulation;
Quantify the total air mass transport along (adiabatic) /across (diabatic) an isentropic level in a latitude band;
Single-cell, warm and cold air branch; the stratospheric branch variability is a good indicator of individual CAOs (Yu et al., 2015; Cai et. al .2016).

5. Horizontal resolution Definition and physical meaning
Data and Indices
Dataset
Variables
Horizontal resolution
Vertical resolution
Time span
ERA-Interim
T, , Ps, SAT, , etc.
1.5°× 1.5°
37 pressure levels
（1000hPa-1hPa）
Indices
Definition and physical meaning
MMC Indices
PSM
Total air mass in the polar region (60-90°N) above 400K isentrope.
AMT
Total air mass transported adiabatically into the polar region above 400K isentrope.
DMT
Total air mass transported diabatically out of the polar stratosphere across 400K isentrope.
Wave Indices WA
Equivalent amplitude of total waves at each level at 60°N. WT
Equivalent vertically (westward) tilt angle of total waves at each level at 60°N.
The data we use is ERA_Interim with a time span from 1979 to 2011

6. Mass budget analysis: linking the NAM with Meridional Mass Circulation Variability
Adiabatic Mass Transport into the polar stratosphere (AMT)
Polar Stratospheric Mass (PSM)
Auto-correlation
AMT Spectrum
DMT Spectrum
自相关系数
频率谱
OK, now Let us begin the mass budget analysis for the polar stratosphere.
Consider it is the polar stratosphere above 400 K.
PSM is almost the synonym of negative NAM, high negative cor, decorrelation time of 3-4 weeks and a red-noise like spectrum.
Then we examine the timescale of the two terms and found that DMT
慢过程，快过程，增长过程是慢。How does these processes at different timescales result in the low freq var of NAM?
Diabatic Mass Transport into the polar troposphere (DMT)
Mass budget equation over the polar stratosphere

7. Mass budget analysis: linking the NAM with Meridional Mass Circulation Variability
The DMT
de-correlation timescale of 3–4 weeks;
a red-noise-like spectrum;
Associated with a larger PSM is a stronger DMT.
is mainly a thermal damping term due to slow radiative cooling processes. 
The AMT
a white-noise-like spectrum (i.e., has both high-frequency and low-frequency variations);
perfectly correlated with the day-to-day tendency of PSM.
What processes give rise to the variability of AMT at different timescales and its relationship with PSM?
For DMT,

8. Mass budget analysis: Roles of Baroclinic Waves
Co-spectra of the AMT with WA at 70 hPa
Auto-correlations of wave indices WT
Co-spectra of the AMT with WT at 70 hPa
According the MMC theory, in mid-lat, waves are the main driving force of the MMC.
Thus, we first examined the auto-correlations

9. Mass budget analysis: Roles of Baroclinic Waves
AMT_L
AMT_H
WA~
WT~
AMT_L
AMT_H
WA~
WT~
Below-normal PSM
Normal PSM
WA increases
Wave-mean flow interaction
WA increases
WA decreases
AMT_L
WA~
WT~
AMT_H
AMT_L
AMT_H
WA~
WT~
Above-normal PSM
Based on those results, We conjecture that the AMT_H might vary as a stochastic forcing, stronger and weaker, alternatively.
while AMT_L is coupled with PSM via slow wave amplitude variations modulated by PSM.
I try to present the process using this schematic figures
Normal PSM
WA decreases

10. Mass budget analysis: Roles of Baroclinic Waves
Lagged correlations with PSM WA WT
Oscillatory-type feedback from slow amplitude vacillations of planetary waves modulated by PSM
~Internal mode
The conjecture has been confirmed by examining xx
Robust 90-out-of-phase relation between AMT_L and WA with PSM
While AMT_H and WT …
High-frequency variability due to variations of vertically westward tilting of planetary waves, which is not robustly related with PSM ~external forcing

11. 90° out-of-phase coherence
A Linear Stochastic Model for the Low-frequency Nature of NAM Variability
In-phase coherence
90° out-of-phase coherence
when PSM lags
Phase differences (units: degree)
AMT and PSM
DMT and PSM
Next we wish to construct a … based on the results above.
Since in-phase-relation exists in DMT and PSM.
the 90-degree out-of-phase relation exists in AMT and PSM
This leads to the equations on the right (you don’t need to say anything about these figures beyond the key point outlined above,
So this would take little time)
α=3.3 × 10−7 s− 1
κ(ω)=2 × 10−7 s−1,υ=0.68<1

12. A Linear Stochastic Model for the Low-frequency Nature of NAM Variability
Amplify the red-noise response to F at all frequencies, which is more pronounced at lower frequencies
Spectrum of PSM:
|1-υ|<1
Red-noise response of PSM to F only
Damp the red noise response to F with the strongest damping effect felt in the lowest frequency
Accordingly, the spectrum
Modification part due to relation among them
Acts in the opposite way.

13. A Linear Stochastic Model for the Low-frequency Nature of NAM Variability
Spectrum of F (solid) and White noise (dashed)
Spectrum of PSM (F)
Spectrum of PSM (W)
Such simplified model is reasonable since the constructed PSM spectrum is almost identical with the observed one.
The function of each term is also as expected:
In addition, since F is white-noise like, we substitute it with a pure white noise, the results remains the same.
black curve: Observation;
light blue curve: response to F only;
blue curve: the response to F with thermal damping included;
purple curve: the response to F with a net in-phase feedback between the AMT and PSM;
red curve: the response to F with both in-phase and 90° out-of-phase feedbacks between the AMT and PSM.

14. A Linear Stochastic Model for the Low-frequency Nature of NAM Variability
Finally, the low-frequency nature of PSM can be modeled as a linear stochastic model with a low-frequency amplification response of PSM/NAM to a stochastic forcing.
Till now, questions still remain:

15. Take-home Message
-NAM~PSM-->Mass budget equation over the polar stratosphere directly explains the dynamical and thermal processes contributing to the low- frequency variability in polar pressure/height associated with NAM;
The low-frequency nature of the stratospheric NAM is an amplified red- noise response
to a stochastic forcing from the westward tilting variability of waves (AMT_HighFreq)
with a low-frequency amplification feedback from low-frequency amplitude vacillations of planetary waves (AMT_LowFreq)
and thermal damping (DMT).
To sum up,

16. THANKS!
This work has been published on Climate Dynamics (2017) named as “A stochastic model with a low-frequency amplification feedback for the stratospheric northern annular mode”.
WCRP/SPARC Workshop 2017
Seoul, South Korea

17. Mass budget analysis: linking the NAM with Meridional Mass Circulation Variability
Lagged correlations of various mass circulation
indices with (a) the AMT and (b) the DMT
In-phase coherence
90° out-of-phase coherence
when PSM lags
Phase differences (units: degree)
AMT and PSM
DMT and PSM
Next we examined the lagged cor… and the in-phase, 90-out-of-phase correlation of AMT, DMT with PSM at each freq domain
Two main features:
1) AMT & PSM tendency:
perfect correlation, PSM of the same sign follows;
Such large cor is found when AMT leads PSM by ¼ cycle at all frequencies;
in-phase cor is small though slightly larger at low freq.
phase diff=90.
2) DMT&PSM:
Positive cor, but contribute little to PSM tendency. Positive cor is mainly at low freq.

18. Roles of Baroclinic Waves