# The Sensitivity of Characteristics of Large Scale Baroclinic Unstable Waves in Southern Hemisphere to the Underlying Climate

Authors: Sergei Soldatenko; Chris Tingwell
Journal: Advances in Meteorology
 (2013)

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## Abstract

The sensitivity of the main characteristics of baroclinically unstable waves with respect to fundamental parameters of the atmosphere (the static stability parameterσ0 and vertical shear of a zonal wind Λ) is theoretically explored. Two types of waves are considered: synoptic scale waves and planetary scale (ultralong) waves based on an Eady-type model and model with vertically averaged primitive equations. Sensitivity functions are obtained that estimate the impact of variations
in σ0 and Λ on the growth rate and other characteristics of unstable waves and demonstrate that waves belonging to the short-wave part of the spectrum of unstable waves are more sensitive to changes in the static stability parameter than waves belonging to the long-wave part of the spectrum. The obtained theoretical results show that the increase of the static stability and decrease of the meridional temperature gradient in midlatitude baroclinic zones in some areas of the southern hemisphere lead to a slowing of the growth rate of baroclinic unstable waves and an increasing wavelength of baroclinic unstable wave maximum growth rate, that is, a spectrum shift of unstable waves towards longer wavelengths. These might affect the favorable conditions for the development of baroclinic instability and, therefore, the intensity of cyclone generation activity.


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## Body

## 1. Introduction

Static stability and the meridional temperature gradient (MTG) are among the most important fundamental parameters characterizing the state of the atmosphere and, in particular, midlatitude large-scale eddy dynamics [1, 2]. Static stability and MTG play a significant role in the development of baroclinic instability which is the dominant mechanism for generating large-scale atmospheric eddies (cyclones) that form the storm tracks in midlatitudes. The physical nature of baroclinic instability is well understood and explained in the scientific literature, including text books on dynamic meteorology (e.g., [1–4]). Baroclinic instability can be viewed as sloping convection where growing perturbations draw upon the available potential energy which is proportional to a meridional temperature gradient. Since the publication of the pioneering theoretical works of Charney [5] and Eady [6], in which the fundamental baroclinic mechanism of the atmospheric large-scale instability was first described, many scientific papers have been published that examine the growth of initially infinitesimal perturbations in the atmosphere and ocean caused by baroclinic effects. Both linear theory for the onset of baroclinic instability and its nonlinear saturation have been explored in many research articles. A thorough review of baroclinic instability is presented in [7] and to some extent in [8, 9]. Theoretical models of baroclinic instability typically represent linearized dynamics equations and the instability problem is examined as an eigenvalue problem. The nonmodal instability analysis in both linear and nonlinear formulations is more general than eigenanalysis. This technique suggests the solution of a Cauchy (initial value) problem (e.g., [10–12]). Analytical weakly nonlinear theories of baroclinic instability represent a further extension of research (e.g., [13–16]) which focuses on the finite-amplitude behaviour of unstable baroclinic waves. Other areas of studies have explored the life-cycle behaviour of baroclinic unstable waves to understand how initially infinitesimal perturbations grow to large but finite amplitude modifying the mean flow. This kind of research requires numerical integration of the atmospheric nonlinear model equations (e.g., [17–20]).It is important to underline the significant role of the two-layer model of Philips [21, 22] and its modifications (e.g., [13, 23–28]) in developing baroclinic instability theories. However, for this purpose a two-layer model was usually used in the quasigeostrophic approximation. A generalized baroclinic instability analysis based on two-level primitive-equation model has been described in [29] and applied to detailed consideration of the baroclinic instability mechanism.There is evidence of an increase over the past decades of the static stability in the extratropics [30] and a pole-ward movement of the midlatitude precipitation zones and storm tracks which are uniquely linked to zones with strong MTG-baroclinic zones (e.g., [31–35]). Recent studies have also indicated an increase in intensity of extratropical cyclones and a decrease in frequency [36–44]. Changes in geographical locations and intensity of storm tracks are more distinct in the southern hemisphere (SH) (e.g., [45]) affecting the essential features of weather patterns over large territories such as Australia [46–54] and indicating a change of favourable conditions for baroclinic instability because the key role of baroclinic instability in the development of midlatitude cyclones is well established.To study how baroclinic instability has changed in the SH in recent decades, a comparative analysis of climates for periods 1949–1968 and 1975–1994 has been carried out in [50, 54] using National Centres for Environmental Prediction (NCEP) reanalyses and the European Centre for Medium-Range Weather Forecasting (ECMWF) Reanalysis (ERA40) data.  As a measure of the baroclinic instability the Phillips criterion [55] adapted to spherical geometry was chosen:
(1)u-(300hPa)-u-(700hPa)≥bkcpσ-a0Ωcosφsin2φ,
where u-(300hPa) and u-(700hPa) are the zonal wind velocities at 300 hPa and 700 hPa isobaric levels, respectively, σ- is the static stability measure for a given reference state, cp is the specific heat of air at constant pressure, Ω is the angular rotation speed of the earth, a0 is the earth’s radius, bk is a dimensionless constant, and φ  is a latitude. J. S. Frederiksen and C. S. Frederiksen [50] obtained a significant decreasing trend in baroclinic instability over the middle latitudes of the SH. The most significant negative trend is localized between 30S and 40S. For instance, the difference between the two periods, 1949–1968 and 1975–1994, reached a maximum of 17% in the SH subtropical jet stream (for the July climate). In addition it was found that further poleward the baroclinic instability intensified creating favourable conditions for cyclogenesis. Using a two-level linearized primitive equation model, Frederiksen et al. [51] studied the growth rate of unstable modes for the July reference state averaged over the above mentioned periods. They identified a 30% reduction in the growth rate of cyclone-scale modes between the two periods.It should be pointed out that, since the beginning of the 1980s, exploration of the essential features of the atmospheric general circulation in the SH has been facilitated by the availability of plentiful satellite upper-air data. In particular, the importance of baroclinic instability and large-scale eddies in the formation of zonal wind characteristics, such as a tropospheric double-jet phenomenon, was studied in [56, 57].Taking into account the significant role of baroclinic instability in the development of midlatitude cyclones and the intensity changes of baroclinic instability in some areas of the SH in recent decades, this paper examines the sensitivity of the main characteristics of baroclinically unstable waves (e.g., the growth rates of unstable waves as function of wavelength) to fundamental atmospheric parameters: the static stability parameterσ0 which is characterized by temperature lapse rate Γ=-∂T/∂z,  and zonal wind vertical shear Λ which, by thermal wind balance, characterizes the  meridional temperature gradient. Two classes of waves [55] are considered. The first one is characterized by the Rossby number Ro≡U/(fL)~0.1 and the representative horizontal length scale L which is smaller than the Earth’s radius a0, that is, L/a0~0.1. Here U is a horizontal velocity scale and f is the Coriolis parameter.  For this class of waves ζ≫D, where ζ is the vertical component of relative vorticity and  D  is the horizontal divergence. Synoptic scale waves belong to this type of motion [58]. The second class, which includes planetary, or ultralong, waves [59], is characterized by Ro~0.01 and L/a0~1. For these waves the magnitude of horizontal divergence is comparable with the vertical component of vorticity; that is, ζ≈D. To study synoptic scale waves, the Eady-type model is used with uniform zonal wind shear between upper and lower boundaries on an f-plane. In this context parameters σ0 and Λ are considered to be variables that control the development of baroclinic instability in the atmosphere. Ultralong waves are investigated based on a model with vertically averaged equations [60, 61] with a β-plane approximation. Simplified models such as the Eady model of baroclinic instability and models with vertically averaged equations, despite their simplicity, allow solutions to be obtained that clearly illustrate real physical processes in the atmosphere.


## 2. Synoptic Scale Baroclinically Unstable Waves

### 2.1. The Model Equations

We consider the inviscid primitive equation atmospheric model in normalized isobaric coordinates(x,y,ξ) on  an f-plane in the following form:
(2)(∂∂t+u·∇+ωP0∂∂ξ)u+fk×u=-∇Φ,(∂∂t+u·∇)T-Sξω=Qcp,∇·u+1P0∂ω∂ξ=0,∂Φ∂ξ=-RTξ.
The state variables of the model are the horizontal velocity vector u=(u,v)T, the vertical pressure velocity ω≡dp/dt, where p is pressure, the geopotential Φ, and the temperature T. The operator ∇≡(∂/∂x,∂/∂y) is applied to the horizontal coordinates x and y, directed eastward and northward, respectively. The normalized pressure ξ=p/P0, where P0=1000 hPa is a “standard” pressure approximately equal to the surface pressure, is taken as the vertical coordinate, while the time is denoted by t. Other notations are the diabatic heating rate per unit time per unit mass Q, the gas constant R, the unit vector in the vertical direction k, and the reference state static stability measure Sξ in the normalized isobaric coordinate system:
(3)Sξ=RT-gP0ξ(Γd-Γ),
where T- is a reference temperature, g is the gravity acceleration, Γd is the dry adiabatic lapse rate, and Γ is the reference state lapse rate. We employ the f-plane approximation so that the Coriolis parameter f is a constant f=f0=2Ωsinφ0 with φ0 being the latitude of interest. Hereafter we consider only adiabatic process and thus assume zero heating rate Q. The following boundary conditions are used for the pressure velocity:
(4)ω=0atξ=0,ξ=1.The atmospheric reference state, defined byu-, v-, ω-, T-, and Φ-, is steady and satisfies the following relations:
(5)u-=-1f0∂Φ-∂y,v-=0,ω-=0,∂Φ-∂ξ=-RT-ξ,
where T-=T-(y,ξ). By substituting (5) into the set of (2), one can see that (5) is a solution of (2) that describes the zonal flow:
(6)∂u-∂ξ=Rf0ξ∂T-∂y
which matches the specified distribution of the zonally averaged temperature T-(y,ξ) and represents thermal wind balance.  To consider only the baroclinic mechanism of the atmospheric instability, meridional variability of the basic zonal flow is excluded. In other words the barotropic impact on the instability of the basic zonal flow is not taken into account. Therefore we assume that the velocity of the basic zonal flow does not depend on the horizontal y-coordinate; that is, u-=u-(ξ). Thus, the problem now is the study of the instability of the basic zonal flow (6) with respect to infinitesimal perturbations. For this purpose the system (2) is linearized around the basic state (5). Representing the state variables as ψ(x,y,ξ,t)=ψ-(ξ)+ψ′(x,ξ,t), where ψ- is a basic state and ψ′ is an infinitesimal perturbation, and taking into account the hydrostatic equations and the thermal wind relationship (6), the linearized system can be written as
(7)∂u′∂t+u-∂u∂x+ω′P0∂u-∂ξ=-∂Φ′∂x+f0v′,∂v′∂t+u-∂v′∂x+f0u′=0,∂u′∂x+1P0∂ω′∂ξ=0,∂∂t(∂Φ′∂ξ)+u-∂∂x(∂Φ′∂ξ)-v′f0∂u-∂ξ+σ0P0ω′=0.The static stability parameterσ0 is expressed as
(8)σ0=-αP0∂lnΘ-∂ξ=R2T-gP02ξ2(Γd-Γ),
where α is a specific volume and Θ-=T-ξ-R/cp is a reference potential temperature. Suppose that ∂u-/∂ξ=-Λξ=const. Applying the method of separation of variables, we assume the solutions of the form
(9)ψ′(x,ξ,t)=ψ^(ξ)eik(x-ct),
where ψ^(ξ) is a function of ξ only, k is a wave number, and c is a phase velocity of perturbations which in general is a complex value: c=cr+ici. After substituting (9) into (7), we can finally obtain the following single equation for ω^:
(10)(u--c)[1-k2f02(u--c)(u--c)]∂2ω^∂ξ2+2Λξ∂ω^∂ξ-σ0P02k2f02(u--c)ω^=0.
Similar equations were considered in a number of publications (e.g., [4, 62, 63]). The analytical solution of this equation can be obtained by eliminating gravity waves from the consideration and assuming that σ0=const. For this particular case, (10) can be transformed into
(11)(u--c)∂2ω^∂ξ2+2Λζ∂ω^∂ξ-(u--c)σ0P02k2f02ω^=0.
The boundary conditions for ω^ are specified as
(12)ω^=0atξ=0,ξ=1.
Equation (11), together with boundary conditions (12), represents the eigenvalue problem for the complex phase speed c. The fundamental solutions of (11) are expressed through the Bessel functions of the first and second kinds. With the homogeneous boundary conditions (12), the following two discrete eigenvalues can be obtained [4]:
(13)c1,2=Λξ2[1±1-4η2(ηcoth(η)-1)],
where
(14)η=P0σ0(kf0).
The phase velocity c will amplify exponentially if c has an imaginary part ci. From (13) we can see that this will occur if the discriminant in (13) is less than zero:
(15)4η2(ηcoth(η)-1)>1,
which gives by the Newton’s iteration algorithm the necessary condition for instability: η<ηc≈2.3994. Besides two discrete eigenvalues (13), the eigenvalue problem (11)-(12) has a continuous spectrum of eigenvalues c∈(0,u-(ξ)) that are real and therefore can be neglected in the problem of baroclinic instability [4]. The growth rate of unstable waves χk≡kci is calculated by the following expression:
(16)χk=Λξf02P0σ0|η2-4(ηcoth(η)-1)|.As we can see from (16), at a given latitude the growth rate χk is a function of σ0 (static stability), Λξ (wind shear), and quantity η that depends on the horizontal wavenumber k and σ0.


### 2.2. Impact of Static Stability and Vertical Wind Shear on Baroclinic Instability

Within the Eady problem framework, the static stability parameterσ0 and the vertical wind shear Λξ represent the main control variables. By varying σ0 and Λξ, one can obtain estimates of the impact of these parameters on the development of baroclinic instability in the atmosphere. In this research, parameters corresponding to the basic state are given the following values: Λξ=40 ms−1 [50] and σ0=2×10-6 m2 Pa−2 s−2 [1]. These parameter values can be used as an approximation to describe the zonal-averaged atmospheric conditions for JJA (June, July, and August) in the SH [50]. The latitude of interest is assumed to be φ0=45 S which gives f0=-1.028×10-4 s−1.Figure1 shows plot of Eady growth rate versus zonal wavenumber obtained with (16). The growth rate has a short-wave instability cutoff beyond which waves are stable. Let Lmin be the wavelength that corresponds to a short-wave cutoff. Value of Lmin can be obtained from (14) when  η=ηc which gives Lmin=3592 km. To calculate the wavelength of maximum growth rate Lχmax, one can take ∂χk/∂k and set the result equal to zero which gives η=ηm≈1.6061. Then by using (14) we can obtain Lχmax=5366 km.Figure 1
Growth rateχk versus zonal wavenumber kz for σ0=2×10-6m2 Pa−2 s−2 and Δξ=40 m s−1.The influence of the static stability parameter on the wavelength of maximum growth rateLχmax and the short wave cut-off Lmin are shown in Figure 2. In general, an increase in the parameter σ0 leads to an increase in both Lkcmax and Lmin. The functional dependences between Lmin and σ0, and between Lkcmax and σ0 are almost linear: ±10% departure of static stability parameter Δσ0 from its nominal value σ0=2.0×10-6 m2 Pa−2 s−2  results in about ±5% change for both Lkcmax and Lmin with respect to the nominal value σ0. For instance, if Δσ=0.1×σ0, then Lkcmax=5628 and Lmin=3768 km, and if Δσ=-0.1σ0, then Lkcmax=5091 and Lmin=3408 km.Figure 2
Length of waves with maximum growth ratesLkcmax  and short-wave cutoff Lmin, as functions of the static stability parameter σ0.Figure3 illustrates the growth rate of unstable waves versus the static stability parameter at different values of Λξ. Parameters σ0 and Λξ influence the growth rate χk in the opposite direction: growth rate decreases if σ0 increases and if Λξ decreases. Note that the decrease of the parameter Λξ indicates the weakening of the intensity of the baroclinic zone, that is, reduction of the MTG. In nature both of these processes take place, which leads to a synergistic effect. For instance, if Λξ decreases by 10% and the static stability parameter increases by 10%, the growth rate χk decreases by 14%.Figure 3
Growth rateχk versus static stability parameter σ0 for different values of parameter Λξ (units m s−1).Sinceχk is a nonlinear function of σ0 (16), to estimate the influence of infinitesimal perturbations in σ0 on variations in χk, the sensitivity function
(17)Sσ=∂χk∂σ0
and the relative sensitivity function
(18)SσR=∂χk/χk∂σ0/σ0=σ0χk∂χk∂σ0
can be used. The function Sσ shows changes in χk due to variations in σ0. The relative sensitivity function SσR is used to compare model parameters to find out what parameter is the most important for a certain percent change in the parameters. Sensitivity functions (17) and (18) are evaluated in the vicinity of some nominal value of the parameter σ0. We can select several nominal values to cover some range of changes in σ0. Differentiating (16) with respect to control parameter σ0, we can obtain the expression for Sσ(19)Sσ=χk2σ0(P0σ0kf0η-2coth(η)+2ηcsch2(η)η2-4(ηcoth(η)-1)-1).
Sensitivity Sσ versus zonal wavenumbers for different values of σ0 with Λξ=40 m s−1 are shown in Figure 4. The absolute value of the sensitivity of χk with respect to σ0 exponentially increases with decreasing wavelength. For planetary scale waves (zonal wave numbers 1–4), the absolute value of the sensitivity of χk with respect to σ0 is palpably less than sensitivity for synoptic scale waves (zonal wave numbers ≥5). Absolute and relative sensitivity functions Sσ and SσR calculated for different values of σ0 for various zonal wave numbers are shown in Tables 1 and 2, respectively.Table 1
Absolute sensitivitySσ as a function of zonal wavenumber kz for different values of static stability parameter σ0.
σ
0 m2 Pa−2 s−2
Zonal wave numberkz
1
2
3
4
5
6
7
1
.
0
×
1
0
-
6
−0.0157
−0.1251
−0.4175
−0.9772
−1.8879
−3.2470
−5.2005
1.5
×
1
0
-
6
−0.0157
−0.1245
−0.4139
−0.9675
−1.8794
−3.3014
−5.5927
2.0
×
1
0
-
6
−0.0157
−0.1240
−0.4110
−0.9626
−1.8979
−3.4964
−6.9371
2.5
×
1
0
-
6
−0.0157
−0.1234
−0.4088
−0.9629
−1.9509
−3.9665
−23.0031
3.0
×
1
0
-
6
−0.0157
−0.1230
−0.4071
−0.9691
−2.0540
−5.3057Table 2
Relative sensitivitySσR as a function of zonal wavenumber kz for different values of static stability parameter σ0.
σ
0 m2 Pa−2 s−2
Zonal wave numberkz
1
2
3
4
5
6
7
1
.
0
×
1
0
-
6
−0.0062
−0.0250
−0.0575
−0.1057
−0.1735
−0.2692
−0.4095
1.5
×
1
0
-
6
−0.0093
−0.0379
−0.0881
−0.1656
−0.2837
−0.4749
−0.8371
2.0
×
1
0
-
6
−0.0124
−0.0509
−0.1201
−0.2375
−0.4220
−0.8004
−1.9939
2.5
×
1
0
-
6
−0.0156
−0.0642
−0.1539
−0.3087
−0.6071
−1.4399
−31.4243
3.0
×
1
0
-
6
−0.0187
−0.0777
−0.1898
−0.3974
−0.8759
−3.4322Figure 4
Sensitivity functionSσ versus zonal wavenumber kz for different nominal values of the static stability parameter σ0 (units 106 m2 Pa−2 s−2).The expression for sensitivity functionSΛ can be easily obtained by differentiating (16) with respect to control parameter Λξ:
(20)SΛ=f02P0σ0|η2-4(ηcoth(η)-1)|.The functionSΛ versus zonal wave number kz for different σ0 is plotted in Figure 5. It is clear to see that for a given value of the parameter σ0 the graph of function SΛ(kz) is very much like the classic picture of the growth rates χk versus zonal wavenumber kz [5]. It is interesting that the relative sensitivity function SΛR does not depend on the wavelength (wavenumber) and for all of the unstable waves is equal to unity:
(21)SΛR=∂χk/χk∂Λξ/Λξ=Λξχk∂χk∂Λξ=1.
Since relative sensitivity functions allow direct comparison of the importance of model parameters on the growth rate χk, we can see that because SΛR=1 the parameter Λξ (i.e., the meridional temperature gradient) is more important than the static stability parameter σ0 except for σ0>2×10-6 m2 Pa−2 s−2 for waves with kz≥6 (see Table 2). Wherever the midlatitude values in Table 2 are less than one, the growth rate is more sensitive to the meridional temperature gradient (i.e., Λξ) than the static stability (σ0).Figure 5
Sensitivity functionSΛ versus zonal wavenumber kz for different nominal values of the static stability parameter σ0 (units 106 m2 Pa−2 s−2).The obtained results are consistent with observations [50, 51, 54]: an increase in static stability and a decrease of the MTG have occurred over the past few decades in some areas of the SH, which has led to a decrease in the growth rate of baroclinic unstable waves, a shift of the spectrum of unstable waves in the long wavelength part of spectrum, and a weakened intensity of cyclogenesis. Naturally, these changes affect favourable conditions for the development of baroclinic instability and the essential features of weather patterns over large territories, particularly over Australia.


## 2.1. The Model Equations

We consider the inviscid primitive equation atmospheric model in normalized isobaric coordinates(x,y,ξ) on  an f-plane in the following form:
(2)(∂∂t+u·∇+ωP0∂∂ξ)u+fk×u=-∇Φ,(∂∂t+u·∇)T-Sξω=Qcp,∇·u+1P0∂ω∂ξ=0,∂Φ∂ξ=-RTξ.
The state variables of the model are the horizontal velocity vector u=(u,v)T, the vertical pressure velocity ω≡dp/dt, where p is pressure, the geopotential Φ, and the temperature T. The operator ∇≡(∂/∂x,∂/∂y) is applied to the horizontal coordinates x and y, directed eastward and northward, respectively. The normalized pressure ξ=p/P0, where P0=1000 hPa is a “standard” pressure approximately equal to the surface pressure, is taken as the vertical coordinate, while the time is denoted by t. Other notations are the diabatic heating rate per unit time per unit mass Q, the gas constant R, the unit vector in the vertical direction k, and the reference state static stability measure Sξ in the normalized isobaric coordinate system:
(3)Sξ=RT-gP0ξ(Γd-Γ),
where T- is a reference temperature, g is the gravity acceleration, Γd is the dry adiabatic lapse rate, and Γ is the reference state lapse rate. We employ the f-plane approximation so that the Coriolis parameter f is a constant f=f0=2Ωsinφ0 with φ0 being the latitude of interest. Hereafter we consider only adiabatic process and thus assume zero heating rate Q. The following boundary conditions are used for the pressure velocity:
(4)ω=0atξ=0,ξ=1.The atmospheric reference state, defined byu-, v-, ω-, T-, and Φ-, is steady and satisfies the following relations:
(5)u-=-1f0∂Φ-∂y,v-=0,ω-=0,∂Φ-∂ξ=-RT-ξ,
where T-=T-(y,ξ). By substituting (5) into the set of (2), one can see that (5) is a solution of (2) that describes the zonal flow:
(6)∂u-∂ξ=Rf0ξ∂T-∂y
which matches the specified distribution of the zonally averaged temperature T-(y,ξ) and represents thermal wind balance.  To consider only the baroclinic mechanism of the atmospheric instability, meridional variability of the basic zonal flow is excluded. In other words the barotropic impact on the instability of the basic zonal flow is not taken into account. Therefore we assume that the velocity of the basic zonal flow does not depend on the horizontal y-coordinate; that is, u-=u-(ξ). Thus, the problem now is the study of the instability of the basic zonal flow (6) with respect to infinitesimal perturbations. For this purpose the system (2) is linearized around the basic state (5). Representing the state variables as ψ(x,y,ξ,t)=ψ-(ξ)+ψ′(x,ξ,t), where ψ- is a basic state and ψ′ is an infinitesimal perturbation, and taking into account the hydrostatic equations and the thermal wind relationship (6), the linearized system can be written as
(7)∂u′∂t+u-∂u∂x+ω′P0∂u-∂ξ=-∂Φ′∂x+f0v′,∂v′∂t+u-∂v′∂x+f0u′=0,∂u′∂x+1P0∂ω′∂ξ=0,∂∂t(∂Φ′∂ξ)+u-∂∂x(∂Φ′∂ξ)-v′f0∂u-∂ξ+σ0P0ω′=0.The static stability parameterσ0 is expressed as
(8)σ0=-αP0∂lnΘ-∂ξ=R2T-gP02ξ2(Γd-Γ),
where α is a specific volume and Θ-=T-ξ-R/cp is a reference potential temperature. Suppose that ∂u-/∂ξ=-Λξ=const. Applying the method of separation of variables, we assume the solutions of the form
(9)ψ′(x,ξ,t)=ψ^(ξ)eik(x-ct),
where ψ^(ξ) is a function of ξ only, k is a wave number, and c is a phase velocity of perturbations which in general is a complex value: c=cr+ici. After substituting (9) into (7), we can finally obtain the following single equation for ω^:
(10)(u--c)[1-k2f02(u--c)(u--c)]∂2ω^∂ξ2+2Λξ∂ω^∂ξ-σ0P02k2f02(u--c)ω^=0.
Similar equations were considered in a number of publications (e.g., [4, 62, 63]). The analytical solution of this equation can be obtained by eliminating gravity waves from the consideration and assuming that σ0=const. For this particular case, (10) can be transformed into
(11)(u--c)∂2ω^∂ξ2+2Λζ∂ω^∂ξ-(u--c)σ0P02k2f02ω^=0.
The boundary conditions for ω^ are specified as
(12)ω^=0atξ=0,ξ=1.
Equation (11), together with boundary conditions (12), represents the eigenvalue problem for the complex phase speed c. The fundamental solutions of (11) are expressed through the Bessel functions of the first and second kinds. With the homogeneous boundary conditions (12), the following two discrete eigenvalues can be obtained [4]:
(13)c1,2=Λξ2[1±1-4η2(ηcoth(η)-1)],
where
(14)η=P0σ0(kf0).
The phase velocity c will amplify exponentially if c has an imaginary part ci. From (13) we can see that this will occur if the discriminant in (13) is less than zero:
(15)4η2(ηcoth(η)-1)>1,
which gives by the Newton’s iteration algorithm the necessary condition for instability: η<ηc≈2.3994. Besides two discrete eigenvalues (13), the eigenvalue problem (11)-(12) has a continuous spectrum of eigenvalues c∈(0,u-(ξ)) that are real and therefore can be neglected in the problem of baroclinic instability [4]. The growth rate of unstable waves χk≡kci is calculated by the following expression:
(16)χk=Λξf02P0σ0|η2-4(ηcoth(η)-1)|.As we can see from (16), at a given latitude the growth rate χk is a function of σ0 (static stability), Λξ (wind shear), and quantity η that depends on the horizontal wavenumber k and σ0.


## 2.2. Impact of Static Stability and Vertical Wind Shear on Baroclinic Instability

Within the Eady problem framework, the static stability parameterσ0 and the vertical wind shear Λξ represent the main control variables. By varying σ0 and Λξ, one can obtain estimates of the impact of these parameters on the development of baroclinic instability in the atmosphere. In this research, parameters corresponding to the basic state are given the following values: Λξ=40 ms−1 [50] and σ0=2×10-6 m2 Pa−2 s−2 [1]. These parameter values can be used as an approximation to describe the zonal-averaged atmospheric conditions for JJA (June, July, and August) in the SH [50]. The latitude of interest is assumed to be φ0=45 S which gives f0=-1.028×10-4 s−1.Figure1 shows plot of Eady growth rate versus zonal wavenumber obtained with (16). The growth rate has a short-wave instability cutoff beyond which waves are stable. Let Lmin be the wavelength that corresponds to a short-wave cutoff. Value of Lmin can be obtained from (14) when  η=ηc which gives Lmin=3592 km. To calculate the wavelength of maximum growth rate Lχmax, one can take ∂χk/∂k and set the result equal to zero which gives η=ηm≈1.6061. Then by using (14) we can obtain Lχmax=5366 km.Figure 1
Growth rateχk versus zonal wavenumber kz for σ0=2×10-6m2 Pa−2 s−2 and Δξ=40 m s−1.The influence of the static stability parameter on the wavelength of maximum growth rateLχmax and the short wave cut-off Lmin are shown in Figure 2. In general, an increase in the parameter σ0 leads to an increase in both Lkcmax and Lmin. The functional dependences between Lmin and σ0, and between Lkcmax and σ0 are almost linear: ±10% departure of static stability parameter Δσ0 from its nominal value σ0=2.0×10-6 m2 Pa−2 s−2  results in about ±5% change for both Lkcmax and Lmin with respect to the nominal value σ0. For instance, if Δσ=0.1×σ0, then Lkcmax=5628 and Lmin=3768 km, and if Δσ=-0.1σ0, then Lkcmax=5091 and Lmin=3408 km.Figure 2
Length of waves with maximum growth ratesLkcmax  and short-wave cutoff Lmin, as functions of the static stability parameter σ0.Figure3 illustrates the growth rate of unstable waves versus the static stability parameter at different values of Λξ. Parameters σ0 and Λξ influence the growth rate χk in the opposite direction: growth rate decreases if σ0 increases and if Λξ decreases. Note that the decrease of the parameter Λξ indicates the weakening of the intensity of the baroclinic zone, that is, reduction of the MTG. In nature both of these processes take place, which leads to a synergistic effect. For instance, if Λξ decreases by 10% and the static stability parameter increases by 10%, the growth rate χk decreases by 14%.Figure 3
Growth rateχk versus static stability parameter σ0 for different values of parameter Λξ (units m s−1).Sinceχk is a nonlinear function of σ0 (16), to estimate the influence of infinitesimal perturbations in σ0 on variations in χk, the sensitivity function
(17)Sσ=∂χk∂σ0
and the relative sensitivity function
(18)SσR=∂χk/χk∂σ0/σ0=σ0χk∂χk∂σ0
can be used. The function Sσ shows changes in χk due to variations in σ0. The relative sensitivity function SσR is used to compare model parameters to find out what parameter is the most important for a certain percent change in the parameters. Sensitivity functions (17) and (18) are evaluated in the vicinity of some nominal value of the parameter σ0. We can select several nominal values to cover some range of changes in σ0. Differentiating (16) with respect to control parameter σ0, we can obtain the expression for Sσ(19)Sσ=χk2σ0(P0σ0kf0η-2coth(η)+2ηcsch2(η)η2-4(ηcoth(η)-1)-1).
Sensitivity Sσ versus zonal wavenumbers for different values of σ0 with Λξ=40 m s−1 are shown in Figure 4. The absolute value of the sensitivity of χk with respect to σ0 exponentially increases with decreasing wavelength. For planetary scale waves (zonal wave numbers 1–4), the absolute value of the sensitivity of χk with respect to σ0 is palpably less than sensitivity for synoptic scale waves (zonal wave numbers ≥5). Absolute and relative sensitivity functions Sσ and SσR calculated for different values of σ0 for various zonal wave numbers are shown in Tables 1 and 2, respectively.Table 1
Absolute sensitivitySσ as a function of zonal wavenumber kz for different values of static stability parameter σ0.
σ
0 m2 Pa−2 s−2
Zonal wave numberkz
1
2
3
4
5
6
7
1
.
0
×
1
0
-
6
−0.0157
−0.1251
−0.4175
−0.9772
−1.8879
−3.2470
−5.2005
1.5
×
1
0
-
6
−0.0157
−0.1245
−0.4139
−0.9675
−1.8794
−3.3014
−5.5927
2.0
×
1
0
-
6
−0.0157
−0.1240
−0.4110
−0.9626
−1.8979
−3.4964
−6.9371
2.5
×
1
0
-
6
−0.0157
−0.1234
−0.4088
−0.9629
−1.9509
−3.9665
−23.0031
3.0
×
1
0
-
6
−0.0157
−0.1230
−0.4071
−0.9691
−2.0540
−5.3057Table 2
Relative sensitivitySσR as a function of zonal wavenumber kz for different values of static stability parameter σ0.
σ
0 m2 Pa−2 s−2
Zonal wave numberkz
1
2
3
4
5
6
7
1
.
0
×
1
0
-
6
−0.0062
−0.0250
−0.0575
−0.1057
−0.1735
−0.2692
−0.4095
1.5
×
1
0
-
6
−0.0093
−0.0379
−0.0881
−0.1656
−0.2837
−0.4749
−0.8371
2.0
×
1
0
-
6
−0.0124
−0.0509
−0.1201
−0.2375
−0.4220
−0.8004
−1.9939
2.5
×
1
0
-
6
−0.0156
−0.0642
−0.1539
−0.3087
−0.6071
−1.4399
−31.4243
3.0
×
1
0
-
6
−0.0187
−0.0777
−0.1898
−0.3974
−0.8759
−3.4322Figure 4
Sensitivity functionSσ versus zonal wavenumber kz for different nominal values of the static stability parameter σ0 (units 106 m2 Pa−2 s−2).The expression for sensitivity functionSΛ can be easily obtained by differentiating (16) with respect to control parameter Λξ:
(20)SΛ=f02P0σ0|η2-4(ηcoth(η)-1)|.The functionSΛ versus zonal wave number kz for different σ0 is plotted in Figure 5. It is clear to see that for a given value of the parameter σ0 the graph of function SΛ(kz) is very much like the classic picture of the growth rates χk versus zonal wavenumber kz [5]. It is interesting that the relative sensitivity function SΛR does not depend on the wavelength (wavenumber) and for all of the unstable waves is equal to unity:
(21)SΛR=∂χk/χk∂Λξ/Λξ=Λξχk∂χk∂Λξ=1.
Since relative sensitivity functions allow direct comparison of the importance of model parameters on the growth rate χk, we can see that because SΛR=1 the parameter Λξ (i.e., the meridional temperature gradient) is more important than the static stability parameter σ0 except for σ0>2×10-6 m2 Pa−2 s−2 for waves with kz≥6 (see Table 2). Wherever the midlatitude values in Table 2 are less than one, the growth rate is more sensitive to the meridional temperature gradient (i.e., Λξ) than the static stability (σ0).Figure 5
Sensitivity functionSΛ versus zonal wavenumber kz for different nominal values of the static stability parameter σ0 (units 106 m2 Pa−2 s−2).The obtained results are consistent with observations [50, 51, 54]: an increase in static stability and a decrease of the MTG have occurred over the past few decades in some areas of the SH, which has led to a decrease in the growth rate of baroclinic unstable waves, a shift of the spectrum of unstable waves in the long wavelength part of spectrum, and a weakened intensity of cyclogenesis. Naturally, these changes affect favourable conditions for the development of baroclinic instability and the essential features of weather patterns over large territories, particularly over Australia.


## 3. Planetary Scale Waves

To study the influence of the static stability parameter on the dynamics of planetary scale (ultralong) waves, a thin film approximation is applied. This approximation employs a specific averaging technique over the vertical coordinate to the system of primitive equations [60]. As a result, a two-dimensional set of equations can be obtained that describes the dynamics of a two-dimensional baroclinic film. These equations reproduce all the wavelike solutions that correspond to the main weather-forming modes of three-dimensional models and therefore can be used in theoretical studies of large-scale dynamic processes in the atmosphere. The system of vertically averaged equations can be written as [60]
(22)∂u-∂t+V-·∇u-=fv--Rπ∂∂x(πT-)∂v-∂t+V-·∇v-=-fu--Rπ∂∂y(πT-)∂π∂t+∇·(πV-)=0∂T-∂t+V-·∇T-+RcpT-∇·V-=0,
where π=ps/P0. For instance, if the original primitive equations are written in the Phillips’ vertical coordinate system σ=p/ps [64], the operator for vertical averaging is introduced as ψ-=∫01ψdσ and state variables are represented as ψ=ψ-+ψ′. Equations (22) are obtained by neglecting the orography and terms u′V′¯, v′V′¯,  and T′V′¯ [65]. A detailed linear analysis of the vertically averaged equation (22) is represented in [60]. In particular, two types of wave solutions were found: fast waves that propagate westward and slow waves that move eastward. Within the framework of this model, ultralong waves are always neutral for any vertically averaged zonal wind velocity. Indeed, linearizing (22) around the following basic state:
(23)T0=T0(y),u0=-(Rf0)∂T0∂y,v0=0,π0=1
and assuming the beta plane approximation f=f0+βy, where β=(2Ω/a0)cosφ0, and representing the solution in the form (9), one can finally obtain under different asymptotics the following expressions for four wave solutions [60]: (a)
acoustic waves(24)c1,2=u0±c02+f02k2,(b)
Rossby wave(25)c3=u0-βk2+f02/c02,(c)
baroclinic wave(26)c4=u0-f02u02c02β.Here c02=(1+κ)RT0 and κ=R/cp. These results however are valid only for the specific case of a neutral atmosphere with Γ=Γd [60, 61]. To take into account the atmospheric static stability on the behaviour of ultralong waves the polytropic model of the atmosphere can be used for which
(27)T(x,y,z,t)=T0(x,y,t)-Γ(x,y,t)z,
where T0 is the temperature at the surface and Γ is a vertical temperature gradient. Integrating (27) with respect to vertical coordinate, we can obtain T0=T-(1+RΓ/g) [60]. Assuming the geostrophic approximation on a β-plane, the set of vertically averaged equations can be written as [61]
(28)∂T-∂t+RT-f02α1π(π,T-)+R2T-2gf0α3α2π(π,Γ)-R2T-gf01α2(T-,Γ)+κβR2T-2gf02α4α22∂Γ∂x-βRT-f02(α3+κα1α4α2)1π∂(πT-)∂x-κβRT-f02α3α2π(π,Γ)-R2T-gf0α1α4α2∂T-∂x=0,∂Γ∂t-RT-f0α12α21π(Γ,π)+R2T-2gf0α3α2π(π,Γ)-Rf0α1α2(T-,Γ)-κβRT-f02α1α4α22∂Γ∂x-κβgf02α1α3α4α21π∂(πT-)∂x+κβgf02α12α4α2∂T-∂x=0,∂π∂t-βRf02∂(πT-)∂x=0.
Here α1=1+R(Γ/g), α2=1+2R(Γ/g), α3=R(Γ/g), α4=1-(Γ/Γd), and the operator (A,B)=(∂A/∂x)(∂B/∂y)-(∂A/∂y)(∂B/∂x). The basic state is defined as a stationary solution of system (28) for which
(29)∂T-∂x=0,∂Γ∂x=0,∂π∂x=0,
or in other words
(30)T-=T-0(y),Γ=Γ0(y),π=π0(y).Linearizing (28) around the basic state (30), the following cubic characteristic equation can be obtained in which the second order terms are neglected [61]:
(31)c3+c2{α1α2[λ(α2+3κα4)-u0]}+c{λα1α2[λκα4(2+α3+κα4α2(2α1-1α2))-u0(2+κα4+3α3)(2+α3+κα4α2(2α1-1α2))]}+(λα1α2u02-λ2κα12α4α2u0+2λ3κ2α12α3α42α23)=0,
where λ=βRT-0/f02 and u0=-(R/f0π0)((∂(π0T-0))/∂y).  If the discriminant of this equation is positive, then the wave solution is unstable. The domain of zonal flow instability can be found numerically (see diagram in [61]). In Figure 6, we reproduce only for the 1st quadrant of a Cartesian plane the domain of instability calculated as a function of vertically averaged zonal wind velocity u0 and dimensionless temperature lapse rate Γ/Γd.Figure 6
Domain of instability (filled) as a function of dimensionless temperature lapse rateΓ/Γd and vertically averaged zonal wind velocity u0.The imaginary part of phase velocityci which characterises the growth rate of unstable waves χk≡kci is displayed in Figure 7 as a function of dimensionless temperature lapse rate Γ/Γd for different values of vertically averaged zonal wind velocity u0. A maximum phase velocity ci exists for given values of u0; that is, dependent on the ratio of Γ/Γd. For instance, if u0=20 m s−1 then the maximum value (ci)max≈8.34 m s−1 is reached at Γ/Γd≈0.55. Figure 6 shows that increasing vertically averaged zonal wind u0 is associated with increasing ci. This is further evident in Figure 8, which shows ci as a function of u0 for a range of Γ/Γd values. The lower Γ/Γd and the larger ci, that is, ci, increases with decreasing static stability.Figure 7
Imaginary part of phase speedci versus dimensionless temperature lapse rate Γ/Γd for different values of vertically averaged zonal wind velocity u0.Figure 8
Imaginary part of phase speedci versus vertically averaged zonal wind velocity u0 for different values of dimensionless temperature lapse rate Γ/Γd.


## 4. Concluding Remarks

We have studied theoretically the impact of variations in the static stability parameterσ0 and zonal wind shear Λξ on the characteristics of baroclinically unstable waves of synoptic scales using Eady-type model with the uniform Λξ between upper and lower boundaries on an f-plane. Quantitative estimates of variations in σ0 and Λξ on the growth rate χk, wavelength of maximum growth rate Lχmax, and short-wave cutoff Lmin were obtained.Analytical expressions are derived for sensitivity functions for the growth rateχk with respect to variations in static stability parameter and wind shear velocity. These expressions allow estimating to a first-order approximation the influence of changes in σ0 and Λξ on χk. Analytical expressions for relative sensitivity functions allow estimating the significance of variations in σ0 and Λξ on the growth rate of baroclinically unstable waves with a given zonal wave number.To study the impact of variations in atmospheric static stability and zonal wind velocity on the instability of planetary scale waves, the model with vertically averaged primitive equations withβ-plane approximation was applied. As control parameters, we have used dimensionless temperature lapse rate Γ/Γd and vertically averaged zonal wind velocity u0. We have estimated the influence of Γ/Γd and u0 on the imaginary part of phase speed ci, which was used as a measure of instability.The obtained results are qualitatively consistent with changes in the essential weather patterns that occurred over the last several decades in some areas of the SH and, in particular, over Australia (e.g., [49, 50, 52–54]). Climate change results suggest SH midlatitude static stability σ0 may increase and the MTG (the vertical wind shear Λξ) may decrease, which according to our linear theoretical models leads to a slowing of the growth rate of baroclinic unstable waves χk and an increasing wavelength of baroclinic unstable wave with maximum growth rate Lχmax, that is, a spectrum shift of unstable waves towards longer wavelengths. These might affect the favourable conditions for the development of baroclinic instability and, therefore, the rate of cyclogenesis and a reduction in cyclone intensity. The obtained sensitivity functions demonstrate that waves belonging to the short-wave part of the spectrum of unstable waves are more sensitive to changes in the static stability parameter than waves belonging to the long-wave part of the spectrum.To obtain more realistic estimates of the sensitivity of the growth rate of unstable waves with respect to static stability parameter and MTG, numerical modeling based on a full GCM is required. It is hoped to carry out such work in the future.



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*Source: 981271-2013-12-12.xml*