Stormtrack

 Thomas Schartner$\ast$ Ingo Kirchner Institut für Meteorologie, Freie Universität Berlin

Version from July 28, 2015

$\ast$thomas.schartner@met.fu-berlin.de

This is only a brief documentation about the MiKlip Stormtrack Plugin and is currently still under construction. Comments or any kind of feedback is highly appreciated. Please send an e-mail to the authors.

### 1 Introduction

The 500 mb geopotential height has a very large amount of low-frequency variability. If one were to construct the 500 mb root-mean-square (rms) geopotential height ﬁeld, the result would be dominated by the low-frequency component. Many physically important, high-frequency components, such as those associated with developing baroclinic disturbances would be masked. In order to exhibit the higher frequency variability, one must resort to ﬁltering. The method for ﬁltering geopotential height is the expansion into spherical harmonics for the time scale of less than 10 days.

In section 2, the methods of the calculation procedure are described [Blackmon1976]. Sections 3 and 4 explain the input respectively the output of the stormtrack tool.

### 2 Methods

#### 2.1 Spectral Analysis

As mentioned above, variability can be characterized by the expansion of the 500 mb geopotential height ($Z$) ﬁeld into spherical harmonics. The spherical harmonics ${Y}_{n,m}$ are a complete, orthonormal set of functions with simultaneously satisfy the equations

 ${\nabla }^{2}{Y}_{n,m}=-n\left(n+1\right){Y}_{n,m},$ (1)
 $\frac{{\partial }^{2}}{\partial {\lambda }^{2}}{Y}_{n,m}=-{n}^{2}{Y}_{n,m},$ (2)

where ${\nabla }^{2}$ is the angular part of the Laplacian operator written in latitude and longitude. With a real basis, the conditions of the eigenvalues

 $m=0,1,2,\dots ,\text{integer}$ (3)
 $n=0,1,2,\dots ,\text{integer}$ (4)
 $n\ge m.$ (5)

The functions ${Y}_{n,m}$ are written

 ${Y}_{n,0}^{\left(e\right)}=\left[\right\frac{2n+1}{4\pi }\left]\right{P}_{n}\left(\text{sin}\varphi \right),$ (6)
 ${Y}_{n,m}^{\left(e\right)}=\left[\right\frac{2n+1}{2\pi }\frac{\left(n-m\right)!}{\left(n+m\right)!}{\left]\right}^{\frac{1}{2}}\text{cos}m\lambda {P}_{n}^{m}\left(\text{sin}\varphi \right),$ (7)
 ${Y}_{n,m}^{\left(0\right)}=\left[\right\frac{2n+1}{2\pi }\frac{\left(n-m\right)!}{\left(n+m\right)!}{\left]\right}^{\frac{1}{2}}\text{sin}m\lambda {P}_{n}^{m}\left(\text{sin}\varphi \right),$ (8)

where ${P}_{n}\left(\text{sin}\varphi \right)$ are Legendre polynomials and ${P}_{n}^{m}\left(\text{sin}\varphi \right)$ are associated Legendre functions. The superscripts $\left(e\right)$ and $\left(0\right)$ specify the evenness and oddness of the functions under the substitution $\lambda \to -\lambda$. The eigenvalue $m$ is the longitudinal wavenumber. the diﬀerence $n-m$ gives the number of nodes from pole to pole and also determines the evenness (oddness) of the function ${P}_{n}^{m}\left(\text{sin}\varphi \right)$, for n-m even (odd), under the substitution $\varphi \to -\varphi$. Finally, n is the two-dimensional wavenumber appropriate for the spherical geometry and the basic functions. So, the geopotential height $Z\left(\varphi ,\lambda \right)$ has been expanded in a series of spherical harmonics

 $Z\left(\varphi ,\lambda \right)=\sum _{\begin{array}{c}m=0\\ n\ge m\end{array}}{C}_{n,m}{Y}_{n,m}^{\left(e\right)}+\sum _{\begin{array}{c}m=1\\ n\ge m\end{array}}{S}_{n,m}{Y}_{n,m}^{\left(0\right)}$ (9)

The expansion coeﬃcients ${C}_{n,m}$ and ${S}_{n,m}$ are calculated using the orthogonality property of the spherical harmonics. Every ﬁeld in the data set described above was expanded into a series of spherical harmonics. The series is truncated at

 $m\le n\le N=18.$ (10)

Each ﬁeld is therefore represented by 190 nontrivial expansion coeﬃcients. The complete truncated ﬁeld can be recovered using (9), where the upper limit of summation is $m=n=18$. To construct the ﬁelds the summation is done over only part of the wavenumbers. Regime I is deﬁnied in wavenumber space to be those wavenumbers with

 (11)

Regime I is called as the long waves ort the planetary-scale waves. Regime II is deﬁned by

 (12)

This regime contains the medium-scale waves. Finally, Regime III is deﬁned by

 (13)

This regime contains the short waves. The truncations used to deﬁne the total ﬁeld and the wavenumber regimes are therefore total two-dimensional scale truncations. All three regimes deﬁned above contain planetary-scale longitudinal harmonics $m\lesssim 5$. These waves are distinguished by diﬀerent latitudinal scales and consequently by diﬀerent two-dimensional scales. The choice of the boundaries of the wavenumber regimes is somewhat arbitrary. Neverless, the bulk of the waves in each regime behaves diﬀerently from the bulk of waves in another regime. Each set of expansion coeﬃcients, ${C}_{n,m}\left({t}_{i}\right)$ or ${S}_{n,m}\left({t}_{i}\right)$ for ﬁxed n and m, forms a time series. To display the contributions to these time series comming from diﬀerent frequency domains, a 31-point ﬁlters is used in form of

 ${\overline{C}}_{n,m}={a}_{0}{C}_{n,m}\left({t}_{i}\right)+\sum _{p=1}^{15}{a}_{p}\left[{C}_{n,m}\left({t}_{i+p}\right)+{C}_{n,m}\left({t}_{i-p}\right)\right]$ (14)

to deﬁne new ﬁltered coeﬃcients correspoinding to a low-pass, a medium-pass and a high-pass ﬁlter. The medium-pass ﬁlter is sensitive to frequencies in the period $2.5\lesssim T\lesssim 6\text{days}$. So, the medium-pass ﬁlter is used for the stormtrack acitvies with the calculated coeﬃcients in table 1.

 ${a}_{0}$ 0.2776877534 ${a}_{1}$ 0.1433496840 ${a}_{2}$ -0.1020097578 ${a}_{3}$ -0.1947701551 ${a}_{4}$ -0.0923257264 ${a}_{5}$ 0.0283041151 ${a}_{6}$ 0.0419335015 ${a}_{7}$ 0.0033466748 ${a}_{8}$ 0.0041075557 ${a}_{9}$ 0.0328072034 ${a}_{10}$ 0.0304306715 ${a}_{11}$ -0.0020017146 ${a}_{12}$ -0.0191709641 ${a}_{13}$ -0.0096723016 ${a}_{14}$ -0.0001341773 ${a}_{15}$ -0.0030384857

Table 1: coeﬃcients for data sets with a resolution of 12 hour

The result is the “Standard deviation of bandpassﬁltered Sea Level Pressure anomalies” or the “Standard deviation of bandpassﬁltered Geopotential Height anomalies”.

### 3 Input

The calculation of the stormtrack activity is based on at least 12 hourly geopotential height ﬁeld in 500 hPa or air pressure at sea level (pmsl). Input ﬁelds with a higher temporal resolution than 12 hour (e.g. 6-hourly data) will be rejected.

 Outputdir Output directory mandatory default: /scratch/user/evaluation_system/output/stormtrack Cachedir Cache directory mandatory default: /scratch/user/evaluation_system/cache/stormtrack Cacheclear Option switch to NOT clear the cache. mandatory default: True Variabel geopotential height (zg) or air pressure at sea level (psl). mandatory default: zg Project Choose project, e.g. reanalysis, cmip5, baseline1, baseline0 mandatory Product Choose product, e.g. reanalysis, output mandatory Institute Choose institute of experiment, e.g. MPI-M, ECMWF mandatory Model Choose model of experiment, e.g. MPI-ESM-LR, IFS mandatory Experiment Choose experiment name, e.g. decadal1971, ERAINT mandatory Ensemble Choose ensemble, e.g. r1i1p1, r2i1p1 or ”*” for all members mandatory default: * Firstyear Choose ﬁrst year to be processed. Lastyear Choose last year to be processed. Level Choose level [in Pa], e.g. 50000 only reasonable for zg mandatory default: 50000 Ntask Number of tasks. mandatory default: 24 Accu type Set the accumulation type for Stormtrack - complete, monthly or seasonal (for explanation see text) mandatory default: complete Makepic Set ”True” for make picture with tool movieplotter mandatory default: False Dryrun Set ”True” for just showing the result of ﬁnd_ﬁles and set ”False” to process data. mandatory default: True Caption An additional caption to be displayed with the results

Table 2: Input options for Stormtrack

At ﬁrst, you have to specify your output (Outputdir) and cache (Cachedir) directories. The data paths of input ﬁles can be selected via the typical MiKlip data structure. Choose the Project, Product, Institute, Model and Experiment of the geopotential height ﬁeld or air pressure at sea level you want to process. Further, select ensemble member(s) in the Ensemble operator and specify the variable (Variable) you want to analyze. In Firstyear and Lastyear you can choose the range of years which will be processed. The pressure level(Level) of geopotential height ﬁeld can be chosen. Finally, you have the option to visualize some results (Makepic), to remove the cache directories (Cacheclear), to specify the number of tasks (Ntask) and to show the found input ﬁle(s) from your input parameters based on solr_search (Dryrun). The Accu type deﬁnes the accumulation time. The accumulation time is the time period over which is averaged: complete - the whole time period is averaged, monthly - every month is averaged and seasonal DJF, MAM, JJA and SON is averaged.

### 4 Output

The processed ﬁles can be found in the selected Outputdir. The stormtrack ﬁle contain the stormtrack activity named as ”Standard deviation of bandpassﬁltered Sea Level Pressure anomalies” or as “Standard deviation of bandpassﬁltered Geopotential Height anomalies”. If selected, stormtrack activity is also visualized. Figure 1: Example of Stormtrack activity with a complete accumulation for ERA-int (1979-1982). Figure 2: Example of Stormtrack activity with a monthly accumulation for ERA-int (1979-1982). Figure 3: Example of Stormtrack activity with a seasonal accumulation for ERA-int (1979-1982).

### References

Maurice L Blackmon. A climatological spectral study of the 500 mb geopotential height of the northern hemisphere. Journal of the Atmospheric Sciences, 33(8): 1607–1623, 1976.