# Covariant Lyapunov Vectors of a Quasi-geostrophic Baroclinic Model: Analysis of Instabilities and Feedbacks

Abstract: The classical approach for studying atmospheric variability is based on defining a background state and studying the linear stability of the small fluctuations around such a state. Weakly non-linear theories can be constructed using higher order expansions terms. While these methods have undoubtedly great value for elucidating the relevant physical processes, they are unable to follow the dynamics of a turbulent atmosphere. We provide a first example of extension of the classical stability analysis to a non-linearly evolving quasi-geostrophic flow. The so-called covariant Lyapunov vectors (CLVs) provide a covariant basis describing the directions of exponential expansion and decay of perturbations to the non-linear trajectory of the flow. We use such a formalism to re-examine the basic barotropic and baroclinic processes of the atmosphere with a quasi-geostrophic beta-plane two-layer model in a periodic channel driven by a forced meridional temperature gradient $\Delta T$. We explore three settings of $\Delta T$, representative of relatively weak turbulence, well-developed turbulence, and intermediate conditions. We construct the Lorenz energy cycle for each CLV describing the energy exchanges with the background state. A positive baroclinic conversion rate is a necessary but not sufficient condition of instability. Barotropic instability is present only for few very unstable CLVs for large values of $\Delta T$. Slowly growing and decaying hydrodynamic Lyapunov modes closely mirror the properties of the background flow. Following classical necessary conditions for barotropic/baroclinic instability, we find a clear relationship between the properties of the eddy fluxes of a CLV and its instability. CLVs with positive baroclinic conversion seem to form a set of modes for constructing a reduced model of the atmosphere dynamics.

## Summary (3 min read)

### 1 Introduction

- A classical topic of dynamical meteorology and climate dynamics is the study of mid-latitude atmospheric variability and the investigation of the unstable eddies responsible for the synoptic weather.
- These methods allow for computing CLVs for high dimensional chaotic systems and have led to a renewed interest in the 2 related theory.
- In section 6 the main results are discussed.

### 2 Linear Stability Analysis and Lyapunov Vectors

- Before the authors present the model used in their study (see next section 3), they recapitulate the mathematical background of covariant Lyapunov vectors (CLVs).
- First, they are covariant, hence each element of the basis is a time-dependent solution to the tangent linear equation (see below equation 3).
- The meaning of the BLV can be understood by the following examples.
- Therefore, the cut of the subspaces V −j (t) and V +n−j+1(t) contains only vectors which have an asymptotic grow rate of λj on the interval [−∞,∞].
- The backward iteration leads to a vector which aligns with the fastest growing vector of the backward dynamics in the respective backward subspaces.

### 3 The Model

- This is the first time that CLVs are computed for a geophysical model and then used to characterize the properties of linear stability of its chaotic solutions.
- The model is a spectral version of the classical model introduced by Phillips (1956).
- With the boundary conditions and the adimensionalization, the stream function has the following form in spectral space.
- In their study, the authors are obtaining the CLVs for aperiodic background states over a much longer time period of ca. 25 years.
- The fields show baroclinically unstable eddies moving eastward which can be seen from the phase shift between the upper and lower layer.

### 4 Atmospheric Circulation and the Lorenz Energy Cycle

- The authors study such a decomposition for three different values of the forced meridional temperature gradient ∆T (dotted: 39.81K , dashed: 49.77K, solid: 66.36K).
- All setups feature an unstable stationary solution and an attractor corresponding to a turbulent solution.
- In the lower layer a small eastward jet emerges in the middle of the channel.
- The baroclinic conversion CZP→EP quantifies the exchange between the potential energies of the mean and the eddy fields.
- There are also sinks of energy related to the various terms of friction, diffusion and newtonian cooling.

### 5 The Lorenz Energy Cycle Of The CLVs

- Classical stability analysis interprets the growth of the normal modes by introducing a Lorenz energy cycle between the normal modes and the zonal background state (see, e.g. (Holton, 2004)).
- In their model, this role is played by the total energy Etot.
- The average growth rates of the CLVs measured in these or any other norms is equal to the Lyapunov exponents.
- After applying integration by parts, this term equals the negative correlation between the convergence of heat transport of the CLV (−∂y(v′Mψ′T ) − ∂x(u′Mψ′T )) and the temperature of the background state.

### 6 Physical Properties Of The CLVs

- In this section the authors present the actual features of the LEC and the associated transports for the CLVs.
- Additionally, the authors show the properties of chaoticity that can be derived from the Lyapunov spectrum.

### 6.2.1 Energy Conversion Terms and Sinks

- Now the authors can unravel the connection of the physics of the CLVs captured by the LEC to their stability properties.
- This includes all growing CLVs and part of the decaying CLVs.
- Hence, with increasing ∆T the barotropic conversion of all unstable CLVs turns from negative and to positive values for the fast growing CLVs.
- The energy loss rate by Ekman friction is large for very stable CLVs (low Lyapunov exponents).
- The authors compare the slow growing/decaying CLVs with the decomposition of the flow into zonal mean and the eddies (see section 4) in the magnified view on the right side of the figures 6 and 7.

### 6.2.2 Convergence of Heat and Momentum Transport and Vertical Velocity

- The previously discussed energy conversion (CBC , CBT and CPK) shown in fig.
- For lower ∆T the convergence of heat transport is largest near the center of the channel.
- Correspondingly, the momentum transports of the CLVs cause a convergence of momentum in the center of the channel, resulting a pointier jet (see fig. 8 a).
- This results into the fact that the authors do observe a pointy jet as mean state of the system (see also fig.
- The CLVs with positive baroclinic conversion feature upward and northward heat transport, while the opposite holds for the CLVs with negative baroclinic conversion rate.

### 7 Explaining the variability of the background flow

- So far the authors have studied the linear stability of their model by determining physical properties of CLVs (see section 6).
- Moreover, the Eady modes grow similar to cyclones due to a vertical westward tilt of the troughs and lows which induces a positive baroclinic energy transfer to the modes from the respective zonal background state and a northward eddy heat transport.
- The authors address first how these subspaces of CLVs are constructed.
- In this figure the authors also compare the Bsn cascade with a randomly chosen basis.

### 8 Summary & Conclusion

- The authors objective in this study was to determine the physical properties of the tangent linear space of a quasi-geostrophic model of the mid latitudes atmosphere.
- This understanding of the dynamics is linked to the decomposition of the atmospheric flow into a zonal mean state and an eddy field.
- The slowly growing and decaying CLVs exhibit similar properties as they all feature terms of a positive baroclinic conversion and a negative barotropic conversion including the associated momentum transport to the middle of the channel and the northward heat transport.
- This is due to the slow decorrelation of the slow growing/decaying CLVs with the background trajectory.
- Furthermore, the CLVs will be constructed for models featuring time-scale separations like the authors expect them in the primitive equations and coupled atmosphere/ocean models.

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##### Frequently Asked Questions (2)

###### Q2. What are the future works mentioned in the paper "Covariant lyapunov vectors of a quasi- geostrophic baroclinic model: analysis of instabilities and feedbacks" ?

Future work will deal with two layer QG dynamics with different imposed boundary conditions. It is also promising to study higher resolutions given the discovery of wave-dynamical and damped-advective Floquet vectors by Wolfe and Samelson in a high resolution QG two layer model ( Wolfe and Samelson, 2006, 2008 ). This will include on the one hand orography acting on the lower layer ( Speranza et al., 1985 ; Charney and DeVore, 1979 ) and on the other hand potential vorticity anomalies imposed on the upper layer. The authors will also test the hypothesis whether the CLVs can be used for the construction of a reduced model of the atmospheric circulation.