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

Abstract

The sensitivity of the main characteristics of baroclinically unstable waves with respect to fundamental parameters of the atmosphere (the static stability parameter 𝜎₀ 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.

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: 𝑢^(300hPa)−𝑢^(700hPa)≥𝑏_𝑘𝑐_𝑝𝜎𝑎₀Ωcos𝜑sin²𝜑, (1) 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, 𝑐_𝑝 is the specific heat of air at constant pressure, Ω is the angular rotation speed of the earth, 𝑎₀ is the earth’s radius, 𝑏_𝑘 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 parameter 𝜎₀ 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: 𝜕𝜔𝜕𝑡+𝐮⋅∇+𝑃₀𝜕𝜕𝜕𝜉𝐮+𝑓𝐤×𝐮=−∇Φ,𝜕𝑡+𝐮⋅∇𝑇−𝑆_𝜉𝑄𝜔=𝑐_𝑝,1∇⋅𝐮+𝑃₀𝜕𝜔𝜕𝜉=0,𝜕Φ𝜕𝜉=−𝑅𝑇𝜉. (2) 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 𝑆_𝜉 in the normalized isobaric coordinate system: 𝑆_𝜉=𝑅𝑇𝑔𝑃₀𝜉Γ_𝑑−Γ, (3) where 𝑇 is a reference temperature, 𝑔 is the gravity acceleration, Γ_𝑑 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: 𝜔=0at𝜉=0,𝜉=1. (4)

The atmospheric reference state, defined by 𝑢, 𝑣, 𝜔, 𝑇, and Φ, is steady and satisfies the following relations: 1𝑢=−𝑓₀𝜕Φ,𝜕𝑦𝑣=0,𝜕𝜔=0,Φ𝑅𝜕𝜉=−𝑇𝜉, (5) 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: 𝜕𝑢=𝑅𝜕𝜉𝑓₀𝜉𝜕𝑇𝜕𝑦 (6) 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 𝜕𝑢^′+𝜕𝑡𝑢𝜕𝑢+𝜔𝜕𝑥𝑃^′₀𝜕𝑢𝜕𝜉=−𝜕Φ^′𝜕𝑥+𝑓₀𝑣^′,𝜕𝑣^′+𝜕𝑡𝑢𝜕𝑣^′𝜕𝑥+𝑓₀𝑢^′=0,𝜕𝑢^′+1𝜕𝑥𝑃₀𝜕𝜔^′𝜕𝜕𝜉=0,𝜕𝑡𝜕Φ^′+𝜕𝜉𝑢𝜕𝜕𝑥𝜕Φ^′𝜕𝜉−𝑣^′𝑓₀𝜕𝑢𝜕𝜉+𝜎₀𝑃₀𝜔^′=0. (7)

The static stability parameter 𝜎₀ is expressed as 𝜎₀𝛼=−𝑃₀𝜕lnΘ=𝑅𝜕𝜉²𝑇𝑔𝑃²₀𝜉²Γ_𝑑−Γ, (8) where 𝛼 is a specific volume and Θ=𝑇𝜉^{−𝑅/𝑐𝑝} is a reference potential temperature. Suppose that 𝜕𝑢/𝜕𝜉=−Λ_𝜉=const. Applying the method of separation of variables, we assume the solutions of the form 𝜓^′(𝑥,𝜉,𝑡)=𝜓(𝜉)𝑒^{𝑖𝑘(𝑥−𝑐𝑡)}, (9) 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=𝑐_𝑟+𝑖𝑐_𝑖. After substituting (9) into (7), we can finally obtain the following single equation for 𝜔: (𝑘𝑢−𝑐)1−²𝑓²₀(𝑢−𝑐)(𝜕𝑢−𝑐)²𝜔𝜕𝜉²+2Λ_𝜉𝜕𝜔𝜕𝜉−𝜎₀𝑃²₀𝑘²𝑓²₀(𝑢−𝑐)𝜔=0. (10) 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 ()𝜕𝑢−𝑐²𝜔𝜕𝜉²+2Λ_𝜁𝜕𝜔−(𝜕𝜉𝑢−𝑐)𝜎₀𝑃²₀𝑘²𝑓²₀𝜔=0. (11) The boundary conditions for 𝜔 are specified as 𝜔=0at𝜉=0,𝜉=1. (12) 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]: 𝑐_1,2=Λ_𝜉21±41−𝜂²𝜂coth𝜂−1, (13) where 𝜂=𝑃₀√𝜎₀𝑘𝑓₀. (14) The phase velocity 𝑐 will amplify exponentially if 𝑐 has an imaginary part 𝑐_𝑖. From (13) we can see that this will occur if the discriminant in (13) is less than zero: 4𝜂²𝜂coth𝜂−1>1, (15) which gives by the Newton’s iteration algorithm the necessary condition for instability: 𝜂<𝜂_𝑐≈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 𝜒_𝑘≡𝑘𝑐_𝑖 is calculated by the following expression: 𝜒_𝑘=Λ_𝜉𝑓₀2𝑃₀√𝜎₀𝜂²−4𝜂coth𝜂−1. (16)

As we can see from (16), at a given latitude the growth rate 𝜒_𝑘 is a function of 𝜎₀ (static stability), Λ_𝜉 (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 parameter 𝜎₀ and the vertical wind shear Λ_𝜉 represent the main control variables. By varying 𝜎₀ and Λ_𝜉, 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: Λ_𝜉=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.

Figure 1 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 𝐿_min be the wavelength that corresponds to a short-wave cutoff. Value of 𝐿_min can be obtained from (14) when 𝜂=𝜂_𝑐 which gives 𝐿_min=3592 km. To calculate the wavelength of maximum growth rate 𝐿_𝜒max, one can take 𝜕𝜒_𝑘/𝜕𝑘 and set the result equal to zero which gives 𝜂=𝜂_𝑚≈1.6061. Then by using (14) we can obtain 𝐿_𝜒max=5366 km.

Figure 1 

Growth rate 𝜒𝑘 versus zonal wavenumber 𝑘𝑧 for 𝜎0=2×10−6  m2 Pa−2 s−2 and Δ𝜉=40 m s−1.

The influence of the static stability parameter on the wavelength of maximum growth rate 𝐿_𝜒max and the short wave cut-off 𝐿_min are shown in Figure 2. In general, an increase in the parameter 𝜎₀ leads to an increase in both 𝐿_𝑘𝑐max and 𝐿_min. The functional dependences between 𝐿_min and 𝜎₀, and between 𝐿_𝑘𝑐max 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 𝐿_𝑘𝑐max and 𝐿_min with respect to the nominal value 𝜎₀. For instance, if Δ𝜎=0.1×𝜎₀, then 𝐿_𝑘𝑐max=5628 and 𝐿_min=3768 km, and if Δ𝜎=−0.1𝜎₀, then 𝐿_𝑘𝑐max=5091 and 𝐿_min=3408 km.

Figure 2 

Length of waves with maximum growth rates 𝐿𝑘𝑐max and short-wave cutoff 𝐿min, as functions of the static stability parameter 𝜎0.

Figure 3 illustrates the growth rate of unstable waves versus the static stability parameter at different values of Λ_𝜉. Parameters 𝜎₀ and Λ_𝜉 influence the growth rate 𝜒_𝑘 in the opposite direction: growth rate decreases if 𝜎₀ increases and if Λ_𝜉 decreases. Note that the decrease of the parameter Λ_𝜉 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 Λ_𝜉 decreases by 10% and the static stability parameter increases by 10%, the growth rate 𝜒_𝑘 decreases by 14%.

Figure 3 

Growth rate 𝜒𝑘 versus static stability parameter 𝜎0 for different values of parameter Λ𝜉 (units m s−1).

Since 𝜒_𝑘 is a nonlinear function of 𝜎₀ (16), to estimate the influence of infinitesimal perturbations in 𝜎₀ on variations in 𝜒_𝑘, the sensitivity function 𝑆_𝜎=𝜕𝜒_𝑘𝜕𝜎₀ (17) and the relative sensitivity function 𝑆^𝑅_𝜎=𝜕𝜒_𝑘/𝜒_𝑘𝜕𝜎₀/𝜎₀=𝜎₀𝜒_𝑘𝜕𝜒_𝑘𝜕𝜎₀ (18) can be used. The function 𝑆_𝜎 shows changes in 𝜒_𝑘 due to variations in 𝜎₀. The relative sensitivity function 𝑆^𝑅_𝜎 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 𝑆_𝜎𝑆_𝜎=𝜒_𝑘2𝜎₀𝑃₀√𝜎₀𝑘𝑓₀𝜂−2coth𝜂+2𝜂csch²𝜂𝜂²𝜂.−4𝜂coth−1−1 (19) Sensitivity 𝑆_𝜎 versus zonal wavenumbers for different values of 𝜎₀ with Λ_𝜉=40 m s⁻¹ are shown in Figure 4. The absolute value of the sensitivity of 𝜒_𝑘 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 𝜒_𝑘 with respect to 𝜎₀ is palpably less than sensitivity for synoptic scale waves (zonal wave numbers ≥5). Absolute and relative sensitivity functions 𝑆_𝜎 and 𝑆^𝑅_𝜎 calculated for different values of 𝜎₀ for various zonal wave numbers are shown in Tables 1 and 2, respectively.

Table 1 

Absolute sensitivity 𝑆_𝜎 as a function of zonal wavenumber 𝑘_𝑧 for different values of static stability parameter 𝜎₀.

Table 2 

Relative sensitivity 𝑆^𝑅_𝜎 as a function of zonal wavenumber 𝑘_𝑧 for different values of static stability parameter 𝜎₀.

Figure 4 

Sensitivity function 𝑆𝜎 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 function 𝑆_Λ can be easily obtained by differentiating (16) with respect to control parameter Λ_𝜉: 𝑆_Λ=𝑓₀2𝑃₀√𝜎₀𝜂²−4𝜂coth𝜂−1. (20)

The function 𝑆_Λ versus zonal wave number 𝑘_𝑧 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 𝑆_Λ(𝑘_𝑧) is very much like the classic picture of the growth rates 𝜒_𝑘 versus zonal wavenumber 𝑘_𝑧 [5]. It is interesting that the relative sensitivity function 𝑆^𝑅_Λ does not depend on the wavelength (wavenumber) and for all of the unstable waves is equal to unity: 𝑆^𝑅_Λ=𝜕𝜒_𝑘/𝜒_𝑘𝜕Λ_𝜉/Λ_𝜉=Λ_𝜉𝜒_𝑘𝜕𝜒_𝑘𝜕Λ_𝜉=1. (21) Since relative sensitivity functions allow direct comparison of the importance of model parameters on the growth rate 𝜒_𝑘, we can see that because 𝑆^𝑅_Λ=1 the parameter Λ_𝜉 (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 𝑘_𝑧≥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., Λ_𝜉) than the static stability (𝜎₀).

Figure 5 

Sensitivity function 𝑆Λ 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] 𝜕𝑢+𝜕𝑡𝐕⋅∇𝑢=𝑓𝑅𝑣−𝜋𝜕𝜋𝜕𝑥𝑇𝜕𝑣+𝜕𝑡𝐕⋅∇𝑣=−𝑓𝑅𝑢−𝜋𝜕𝜋𝜕𝑦𝑇𝜕𝜋𝜋𝜕𝑡+∇⋅𝐕𝜕=0𝑇+𝜕𝑡𝐕⋅∇𝑅𝑇+𝑐_𝑝𝑇∇⋅𝐕=0, (22) where 𝜋=𝑝_𝑠/𝑃₀. For instance, if the original primitive equations are written in the Phillips’ vertical coordinate system 𝜎=𝑝/𝑝_𝑠 [64], the operator for vertical averaging is introduced as 𝜓=∫¹₀𝜓𝑑𝜎 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: 𝑇₀=𝑇₀𝑦,𝑢₀𝑅=−𝑓₀𝜕𝑇₀𝜕𝑦,𝑣₀=0,𝜋₀=1 (23) 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 𝑐_1,2=𝑢₀±𝑐²₀+𝑓²₀𝑘², (24)(b)Rossby wave 𝑐₃=𝑢₀−𝛽𝑘²+𝑓²₀/𝑐²₀, (25)(c)baroclinic wave 𝑐₄=𝑢₀−𝑓²₀𝑢²₀𝑐²₀𝛽. (26)Here 𝑐²₀=(1+𝜅)𝑅𝑇₀ and 𝜅=𝑅/𝑐_𝑝. These results however are valid only for the specific case of a neutral atmosphere with Γ=Γ_𝑑 [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 𝑇𝑥,𝑦,𝑧,𝑡=𝑇₀𝑥,𝑦,𝑡−Γ𝑥,𝑦,𝑡𝑧, (27) 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] 𝜕𝑇+𝑅𝜕𝑡𝑇𝑓²₀𝛼₁𝜋𝜋,𝑇+𝑅²𝑇²𝑔𝑓₀𝛼₃𝛼₂𝜋()−𝑅𝜋,Γ²𝑇𝑔𝑓₀1𝛼₂𝑇,Γ+𝜅𝛽𝑅²𝑇²𝑔𝑓²₀𝛼₄𝛼²₂𝜕Γ−𝜕𝑥𝛽𝑅𝑇𝑓²₀𝛼₃𝛼+𝜅¹𝛼₄𝛼₂1𝜋𝜕𝜋𝑇𝜕𝑥−𝜅𝛽𝑅𝑇𝑓²₀𝛼₃𝛼₂𝜋()−𝑅𝜋,Γ²𝑇𝑔𝑓₀𝛼₁𝛼₄𝛼₂𝜕𝑇𝜕𝑥=0,𝜕Γ−𝑅𝜕𝑡𝑇𝑓₀𝛼²₁𝛼₂1𝜋()+𝑅Γ,𝜋²𝑇²𝑔𝑓₀𝛼₃𝛼₂𝜋()−𝑅𝜋,Γ𝑓₀𝛼₁𝛼₂𝑇,Γ−𝜅𝛽𝑅𝑇𝑓²₀𝛼₁𝛼₄𝛼²₂𝜕Γ𝜕𝑥−𝜅𝛽𝑔𝑓²₀𝛼₁𝛼₃𝛼₄𝛼₂1𝜋𝜕𝜋𝑇𝜕𝑥+𝜅𝛽𝑔𝑓²₀𝛼²₁𝛼₄𝛼₂𝜕𝑇𝜕𝑥=0,𝜕𝜋−𝜕𝑡𝛽𝑅𝑓²₀𝜕𝜋𝑇𝜕𝑥=0. (28) Here 𝛼₁=1+𝑅(Γ/𝑔), 𝛼₂=1+2𝑅(Γ/𝑔), 𝛼₃=𝑅(Γ/𝑔), 𝛼₄=1−(Γ/Γ_𝑑), 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 𝜕𝑇𝜕𝑥=0,𝜕Γ𝜕𝑥=0,𝜕𝜋𝜕𝑥=0, (29) or in other words 𝑇=𝑇₀𝑦,Γ=Γ₀𝑦,𝜋=𝜋₀𝑦. (30)

Linearizing (28) around the basic state (30), the following cubic characteristic equation can be obtained in which the second order terms are neglected [61]: 𝑐³+𝑐²𝛼₁𝛼₂𝜆𝛼₂+3𝜅𝛼₄−𝑢₀𝜆𝛼+𝑐¹𝛼₂𝜆𝜅𝛼₄2+𝛼₃𝛼+𝜅⁴𝛼₂2𝛼₁−1𝛼₂−𝑢₀2+𝜅𝛼₄+3𝛼₃+𝜆𝛼₁𝛼₂𝑢²₀−𝜆²𝜅𝛼²₁𝛼₄𝛼₂𝑢₀+2𝜆³𝜅²𝛼²₁𝛼₃𝛼²₄𝛼³₂=0, (31) where 𝜆=𝛽𝑅𝑇₀/𝑓²₀ and 𝑢₀=−(𝑅/𝑓₀𝜋₀)((𝜕(𝜋₀𝑇₀))/𝜕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/Γ_𝑑.

Figure 6 

Domain of instability (filled) as a function of dimensionless temperature lapse rate Γ/Γ𝑑 and vertically averaged zonal wind velocity 𝑢0.

The imaginary part of phase velocity 𝑐_𝑖 which characterises the growth rate of unstable waves 𝜒_𝑘≡𝑘𝑐_𝑖 is displayed in Figure 7 as a function of dimensionless temperature lapse rate Γ/Γ_𝑑 for different values of vertically averaged zonal wind velocity 𝑢₀. A maximum phase velocity 𝑐_𝑖 exists for given values of 𝑢₀; that is, dependent on the ratio of Γ/Γ_𝑑. For instance, if 𝑢₀=20 m s⁻¹ then the maximum value (𝑐_𝑖)_max≈8.34 m s⁻¹ is reached at Γ/Γ_𝑑≈0.55. Figure 6 shows that increasing vertically averaged zonal wind 𝑢₀ is associated with increasing 𝑐_𝑖. This is further evident in Figure 8, which shows 𝑐_𝑖 as a function of 𝑢₀ for a range of Γ/Γ_𝑑 values. The lower Γ/Γ_𝑑 and the larger 𝑐_𝑖, that is, 𝑐_𝑖, increases with decreasing static stability.

Figure 7 

Imaginary part of phase speed 𝑐𝑖 versus dimensionless temperature lapse rate Γ/Γ𝑑 for different values of vertically averaged zonal wind velocity 𝑢0.

Figure 8 

Imaginary part of phase speed 𝑐𝑖 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 parameter 𝜎₀ and zonal wind shear Λ_𝜉 on the characteristics of baroclinically unstable waves of synoptic scales using Eady-type model with the uniform Λ_𝜉 between upper and lower boundaries on an 𝑓-plane. Quantitative estimates of variations in 𝜎₀ and Λ_𝜉 on the growth rate 𝜒_𝑘, wavelength of maximum growth rate 𝐿_𝜒max, and short-wave cutoff 𝐿_min were obtained.

Analytical expressions are derived for sensitivity functions for the growth rate 𝜒_𝑘 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 Λ_𝜉 on 𝜒_𝑘. Analytical expressions for relative sensitivity functions allow estimating the significance of variations in 𝜎₀ and Λ_𝜉 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 with 𝛽-plane approximation was applied. As control parameters, we have used dimensionless temperature lapse rate Γ/Γ_𝑑 and vertically averaged zonal wind velocity 𝑢₀. We have estimated the influence of Γ/Γ_𝑑 and 𝑢₀ on the imaginary part of phase speed 𝑐_𝑖, 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 Λ_𝜉) may decrease, which according to our linear theoretical models leads to a slowing of the growth rate of baroclinic unstable waves 𝜒_𝑘 and an increasing wavelength of baroclinic unstable wave with maximum growth rate 𝐿_𝜒max, 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors thank Dr. I. Pisnichenko for clarification of certain questions relevant to this paper.