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Advances in Meteorology
Volume 2013 (2013), Article ID 981271, 11 pages
The Sensitivity of Characteristics of Large Scale Baroclinic Unstable Waves in Southern Hemisphere to the Underlying Climate
Centre for Australian Weather and Climate Research, 700 Collins Street, Docklands, VIC 3008, Australia
Received 26 September 2013; Accepted 18 November 2013
Academic Editor: Luis Gimeno
Copyright © 2013 Sergei Soldatenko and Chris Tingwell. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The sensitivity of the main characteristics of baroclinically unstable waves with respect to fundamental parameters of the atmosphere (the static stability parameter and vertical shear of a zonal wind ) 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 and 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.
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  and Eady , 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  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  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  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., ) 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  adapted to spherical geometry was chosen: where and are the zonal wind velocities at 300 hPa and 700 hPa isobaric levels, respectively, is the static stability measure for a given reference state, is the specific heat of air at constant pressure, is the angular rotation speed of the earth, is the earth’s radius, is a dimensionless constant, and is a latitude. J. S. Frederiksen and C. S. Frederiksen  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.  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 which is characterized by temperature lapse rate , and zonal wind vertical shear which, by thermal wind balance, characterizes the meridional temperature gradient. Two classes of waves  are considered. The first one is characterized by the Rossby number and the representative horizontal length scale which is smaller than the Earth’s radius , that is, . Here is a horizontal velocity scale and is the Coriolis parameter. For this class of waves , where is the vertical component of relative vorticity and is the horizontal divergence. Synoptic scale waves belong to this type of motion . The second class, which includes planetary, or ultralong, waves , is characterized by and . For these waves the magnitude of horizontal divergence is comparable with the vertical component of vorticity; that is, . To study synoptic scale waves, the Eady-type model is used with uniform zonal wind shear between upper and lower boundaries on an -plane. In this context parameters and 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 -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 on an -plane in the following form: The state variables of the model are the horizontal velocity vector , the vertical pressure velocity , where is pressure, the geopotential , and the temperature . The operator is applied to the horizontal coordinates and , directed eastward and northward, respectively. The normalized pressure , where hPa is a “standard” pressure approximately equal to the surface pressure, is taken as the vertical coordinate, while the time is denoted by . Other notations are the diabatic heating rate per unit time per unit mass , the gas constant , the unit vector in the vertical direction , and the reference state static stability measure in the normalized isobaric coordinate system: where is a reference temperature, is the gravity acceleration, is the dry adiabatic lapse rate, and is the reference state lapse rate. We employ the -plane approximation so that the Coriolis parameter is a constant with being the latitude of interest. Hereafter we consider only adiabatic process and thus assume zero heating rate . The following boundary conditions are used for the pressure velocity:
The atmospheric reference state, defined by , , , , and , is steady and satisfies the following relations: where . By substituting (5) into the set of (2), one can see that (5) is a solution of (2) that describes the zonal flow: which matches the specified distribution of the zonally averaged temperature 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 -coordinate; that is, . 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 , where is a basic state and is an infinitesimal perturbation, and taking into account the hydrostatic equations and the thermal wind relationship (6), the linearized system can be written as
The static stability parameter is expressed as where is a specific volume and is a reference potential temperature. Suppose that . Applying the method of separation of variables, we assume the solutions of the form where is a function of only, is a wave number, and is a phase velocity of perturbations which in general is a complex value: . After substituting (9) into (7), we can finally obtain the following single equation for : 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 . For this particular case, (10) can be transformed into The boundary conditions for are specified as Equation (11), together with boundary conditions (12), represents the eigenvalue problem for the complex phase speed . 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 : where The phase velocity will amplify exponentially if has an imaginary part . From (13) we can see that this will occur if the discriminant in (13) is less than zero: which gives by the Newton’s iteration algorithm the necessary condition for instability: . Besides two discrete eigenvalues (13), the eigenvalue problem (11)-(12) has a continuous spectrum of eigenvalues that are real and therefore can be neglected in the problem of baroclinic instability . The growth rate of unstable waves is calculated by the following expression:
As we can see from (16), at a given latitude the growth rate is a function of (static stability), (wind shear), and quantity that depends on the horizontal wavenumber and .
2.2. Impact of Static Stability and Vertical Wind Shear on Baroclinic Instability
Within the Eady problem framework, the static stability parameter and the vertical wind shear represent the main control variables. By varying and , 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: ms−1  and m2 Pa−2 s−2 . 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 . The latitude of interest is assumed to be S which gives 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 be the wavelength that corresponds to a short-wave cutoff. Value of can be obtained from (14) when which gives km. To calculate the wavelength of maximum growth rate , one can take and set the result equal to zero which gives . Then by using (14) we can obtain km.
The influence of the static stability parameter on the wavelength of maximum growth rate and the short wave cut-off are shown in Figure 2. In general, an increase in the parameter leads to an increase in both and . The functional dependences between and , and between and are almost linear: departure of static stability parameter from its nominal value m2 Pa−2 s−2 results in about % change for both and with respect to the nominal value . For instance, if , then and km, and if , then and km.
Figure 3 illustrates the growth rate of unstable waves versus the static stability parameter at different values of . Parameters and influence the growth rate in the opposite direction: growth rate decreases if increases and if decreases. Note that the decrease of the parameter 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 decreases by 10% and the static stability parameter increases by 10%, the growth rate decreases by 14%.
Since is a nonlinear function of (16), to estimate the influence of infinitesimal perturbations in on variations in , the sensitivity function and the relative sensitivity function can be used. The function shows changes in due to variations in . The relative sensitivity function 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 . We can select several nominal values to cover some range of changes in . Differentiating (16) with respect to control parameter , we can obtain the expression for Sensitivity versus zonal wavenumbers for different values of with m s−1 are shown in Figure 4. The absolute value of the sensitivity of with respect to exponentially increases with decreasing wavelength. For planetary scale waves (zonal wave numbers 1–4), the absolute value of the sensitivity of with respect to is palpably less than sensitivity for synoptic scale waves (zonal wave numbers ≥5). Absolute and relative sensitivity functions and calculated for different values of for various zonal wave numbers are shown in Tables 1 and 2, respectively.
The expression for sensitivity function can be easily obtained by differentiating (16) with respect to control parameter :
The function versus zonal wave number for different is plotted in Figure 5. It is clear to see that for a given value of the parameter the graph of function is very much like the classic picture of the growth rates versus zonal wavenumber . It is interesting that the relative sensitivity function does not depend on the wavelength (wavenumber) and for all of the unstable waves is equal to unity: Since relative sensitivity functions allow direct comparison of the importance of model parameters on the growth rate , we can see that because the parameter (i.e., the meridional temperature gradient) is more important than the static stability parameter except for m2 Pa−2 s−2 for waves with (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., ) than the static stability ().
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 . 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  where . For instance, if the original primitive equations are written in the Phillips’ vertical coordinate system , the operator for vertical averaging is introduced as and state variables are represented as . Equations (22) are obtained by neglecting the orography and terms , , and . A detailed linear analysis of the vertically averaged equation (22) is represented in . 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: and assuming the beta plane approximation , where , and representing the solution in the form (9), one can finally obtain under different asymptotics the following expressions for four wave solutions : (a)acoustic waves (b)Rossby wave (c)baroclinic wave Here and . These results however are valid only for the specific case of a neutral atmosphere with [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 where is the temperature at the surface and is a vertical temperature gradient. Integrating (27) with respect to vertical coordinate, we can obtain . Assuming the geostrophic approximation on a -plane, the set of vertically averaged equations can be written as  Here , , , , and the operator . The basic state is defined as a stationary solution of system (28) for which or in other words
Linearizing (28) around the basic state (30), the following cubic characteristic equation can be obtained in which the second order terms are neglected : where and . 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 ). 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 and dimensionless temperature lapse rate .
The imaginary part of phase velocity which characterises the growth rate of unstable waves is displayed in Figure 7 as a function of dimensionless temperature lapse rate for different values of vertically averaged zonal wind velocity . A maximum phase velocity exists for given values of ; that is, dependent on the ratio of . For instance, if m s−1 then the maximum value m s−1 is reached at . Figure 6 shows that increasing vertically averaged zonal wind is associated with increasing . This is further evident in Figure 8, which shows as a function of for a range of values. The lower and the larger , that is, , increases with decreasing static stability.
4. Concluding Remarks
We have studied theoretically the impact of variations in the static stability parameter and zonal wind shear on the characteristics of baroclinically unstable waves of synoptic scales using Eady-type model with the uniform between upper and lower boundaries on an -plane. Quantitative estimates of variations in and on the growth rate , wavelength of maximum growth rate , and short-wave cutoff were obtained.
Analytical expressions are derived for sensitivity functions for the growth rate 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 and on . Analytical expressions for relative sensitivity functions allow estimating the significance of variations in and 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 -plane approximation was applied. As control parameters, we have used dimensionless temperature lapse rate and vertically averaged zonal wind velocity . We have estimated the influence of and on the imaginary part of phase speed , 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 may increase and the MTG (the vertical wind shear ) may decrease, which according to our linear theoretical models leads to a slowing of the growth rate of baroclinic unstable waves and an increasing wavelength of baroclinic unstable wave with maximum growth rate , 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.
The authors thank Dr. I. Pisnichenko for clarification of certain questions relevant to this paper.
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