c Pleiades Publishing, Ltd., 2009. ISSN 1995-4239, Numerical Analysis and Applications, 2009, Vol. 2, No. 2, pp. 118–130.  c I.V. Borovko, V.N. Krupchatnikov, 2009, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2009, Vol. 12, No. 2, pp. 145–160. Original Russian Text 

The Influence of Stratospheric Polar Vortex Dynamics upon Lower Tropospheric Circulation I. V. Borovko* and V. N. Krupchatnikov** Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch, Russian Academy of Sciences, pr. Akad. Lavrent’eva 6, Novosibirsk, 630090 Russia Received July 10, 2008; in final form, December 3, 2008

Abstract—In this paper, we explore how the extratropical troposphere responds to a polar stratospheric temperature decrease, accompanied by strengthening polar vortex. For this purpose, use is made of a spectral general circulation model with zonally symmetric boundary conditions on surface and a heat source given analytically. DOI: 10.1134/S1995423909020037 Key words: stratosphere, polar vortex, annular mode.

1. INTRODUCTION The stratosphere can affect the conditions of circulation in the lower troposphere in several ways [4–8]. Namely: 1. Ozone concentration variations in the stratosphere affect the ultraviolet radiation flux on the surface and the ozone influx into the lower troposphere. 2. The radiation balance in the lower stratosphere plays an important role in the radiation balance in the troposphere, and concentration variations of greenhouse gases in the lower stratosphere have a direct bearing on the surface temperature. 3. The stratospheric climate system affects weather and climate in the lower troposphere; the existence of this influence has been proved, but its mechanisms are heavily understudied. 4. A signal of solar activity variations propagates to the troposphere from the stratosphere and mesosphere. The influence of solar activity on the weather and climate dynamics in the lower troposphere can probably be explained in the setting of a theory of vertical dynamic stratosphere– troposphere interactions. The stratosphere is a stably stratified medium where potential temperature is monotonically increasing with height. Numerous processes of stratosphere–troposphere interactions can be explained in the framework of a theory of large-scale dynamics [4–7], but small-scale high-frequency processes also have a large influence on the stratospheric dynamics. Many of these processes are produced by Rossby waves, that is, inertial-gravitational waves generated in the troposphere by flows over complex orography, convection, shear instability, etc. Such waves propagate into the stratosphere where they break down and dissipate. This affects the large-scale flow in those stratospheric zones in which waves break down and dissipate. There also exists a nondissipative action owing to horizontal refraction. The breakdown of waves leads to the formation of alternating layers of three-dimensional turbulence where intensive mixing takes place. * **

E-mail: [email protected] E-mail: [email protected]

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2. METHOD To describe atmospheric dynamics, we used a spectral model of atmospheric general circulation [1–3].

2.1. Model The system is solved in coordinates (μ, λ, σ), where μ = sin φ, φ is latitude, λ is longitude, and the vertical coordinate p σ= , (1) ps where p is pressure and ps is surface pressure. The vertical component of absolute vorticity ζ = ξ + f , ∂v ∂u − cos1 φ ∂φ , is taken to be a prognostic variable. where ξ = 1r ∂λ To simulate atmospheric dynamics, we applied the parametrization of radiation heat influxes developed in [2]. Radiative heating in the layer from 30 to 70 km was assumed to be caused by absorption of solar ultraviolet radiation by ozone and emission of infrared radiation by carbon dioxide. A simplified radiative cooling model in which radiative heating depends only on temperature is used for these processes. We have the vorticity equation 1 ∂ ξ ∂ ∂ζ = Fv − Fu − − k(−1)n ∇2n ξ, ∂t 1 − μ2 ∂λ ∂μ τf the divergence equation

 U2 + V 2 D + Φ + TR ln ps − − k(−1)n ∇2n D, 2 2(1 − μ ) τf

(3)

1 ∂ ∂T T ω TR − T ∂ ∂T  =− (uT  ) − (vT  ) + D · T  − σ˙ +κ + − k(−1)n ∇2n T  , 2 ∂t 1 − μ ∂λ ∂μ ∂σ p τR

(4)

1 ∂ ∂ ∂D = Fu + Fv − ∇2 2 ∂t 1 − μ ∂λ ∂μ



(2)

the equation of thermodynamics

the continuity equation U ∂ ln ps ∂ ln ps ∂ σ˙ ∂ ln ps =− −V −D− , 2 ∂t 1 − μ ∂λ ∂μ ∂σ

(5)

and the quasihydrostatic equation ∂Φ = −T. ∂ ln σ The vertical velocity (in terms of ω =

dp dt )

(6)

is found from the equation

ω  · ∇ ln ps − 1 =V p σ

σ  · ∇ ln ps ) dσ, (D + V

(7)

0

 = (u, v); here u is the zonal velocity, v is the meridional velocity, f = 2Ω sin φ is the Coriolis where V parameter, p is pressure, D is the horizontal divergence, T is temperature, and   V = v 1 − μ2 , U = u 1 − μ2 , Fu = V ζ − σ˙

∂U ∂ ln ps − T , ∂σ ∂λ

Fv = −U ζ − σ˙

∂V ∂ ln ps − T  (1 − μ2 ) . ∂σ ∂μ

Here σ˙ is a vertical velocity, T = T0 + T  , Φ is a geopotential, k is a superdiffusion coefficient, n is a superdiffusion order, κ is an adiabatic coefficient, τR = 1/αR is the time scale of radiative cooling, and NUMERICAL ANALYSIS AND APPLICATIONS Vol. 2

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τf is the time scale of Rayleigh friction (assumed to be infinitely large everywhere except the underlying surface). In the equation of thermodynamics, the term TRτR−T is responsible for radiative heating. A radiative equilibrium temperature is written as follows: TR (σ, φ) = Tr (σ) + h(σ), ⎧     1 μ π σ − σT ⎪ 2 ⎪ , σ > σT , ΔTN S − ΔTEP μ − ⎨ sin 2 1 −σT  2 3 h(σ, φ) = σ ⎪ ⎪ , σ < σT . ⎩ ω(φ)ΓH ln σT

(8)

Here Γ is the temperature gradient of radiative equilibrium responsible for the rate of cooling, σT is the coordinate σ of the tropopause, H is the height √ of the stratosphere, and Tr is a standard atmospheric temperature, which depends only on σ, ω = 1 − μ. The system is closed by the following initial and boundary conditions at t = 0: u = u0 (λ, μ, σ), v = v0 (λ, μ, σ), T = T0 (λ, μ, σ), and ps = p0s (λ, μ). At the lower boundary (σ = 1), Φ = Φ0 (λ, μ, σ), σ˙ = 0; at the upper boundary (σ = 0), σ˙ = 0.

2.2. Description of the Numerical Scheme Equations are solved by a grid transformation method. Time derivatives are approximated by a semiimplicit central difference scheme. Variables are represented as a finite sum of spherical functions, i.e., X(λ, μ) =

M



n=N (m) m imλ xm . n Pn (μ)e

m=−M n=|m|

Here X stands for diagnostic variables ξ, D, u, v, T , and ps . In this version of the model, use is made of a triangular truncation of the series by 42 spherical functions, i.e., M = N = 42. In what follows, this spectral resolution is denoted T42. Expansion coefficients xm n are calculated by the Legendre transformation formula 1 m xn = xm (μ)Pnm (μ)dμ, −1

where xm (μ) are the coefficients obtained by using the fast Fourier transform 1 x (μ) = 2π

2π

m

X(μ, λ)e−imλ dλ.

0

The Legendre transformation is calculated on a Gaussian grid with Gaussian quadrature formulas. In this model, prognostic variables are calculated at σ = σk = k−1/2 n , where n is the number of vertical layers (k = 1, n), and the vertical velocity σ˙ is taken at σ = σk+1/2 = nk (k = 0, n − 1). Vertical advection of the quantity X is calculated by the following formula:  Xk+1 − Xk Xk − Xk−1 1 σ˙ + σ˙ k−1/2 , 2 k+1/2 Δσ Δσ where the subscript k is a quantity taken at a level σ = σk . A finite-difference vertical integration analog of Eq. (5) gives the following relations:

∂ ln ps = − Ak Δσk , ∂t n

k=1

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σ˙ k+1/2 =

k



Δσs

n

s=1

As Δσs −

1

k

121

As Δσs ,

s=1

 ∇ ln ps , and where A = D + V Φk − Φk+1 = αk (Tk + Tk+1 ) follows from (6), where αk =

1 2

ln(σk+1 /σk ). The condition at the lower boundary yields Φn − Φ0 = αn Tn ,

where αn = ln(1/σn ). In (4), the term responsible for adiabatic cooling is represented as ⎡ ⎛ ⎞⎤ k k−1



1 ⎝ Aj Δσj + αk−1 Aj Δσj ⎠⎦ Vk · ∇ ln ps − αk κTk ⎣ Δσr j=1

j=1

(see [17]). Applying a finite-difference vertical scheme, we obtain the following system of equations: ∂ζ ∂t ∂D ∂t

(9)

= Z,

= −∇2 (Φ∗ + RGT + RTR ln ps ) + D∗ , ∂T ∂t ∂ ln ps ∂t

= −τ D +

T ∗,

(11)

= −(¯ π · D) − P ∗ .

Here G = (gik ) is a quasihydrostatic matrix, ⎧ ⎪ ⎪ ⎪ ⎨ 0, gik = αi , ⎪ ⎪ ⎪ ⎩α

(10)

(12)

i > k, i = k,

+ αk , i < k,   Δσ X is an n-dimensional vector quantity, τ = (τkj ) = κT0k Gjk Δσkj , Z, T ∗ , D∗ , P ∗ are explicitly treated k−1

sources, and π is a vector (Δσ1 , . . . , Δσn ). Removing the dependent variables from system (9)–(12) yields the following equation for D:   2   ∂D∗ ∂ 2 2 ∗ ∗ − ∇ − B∇ = + T P D GT , R ∂t2 ∂t where the matrix B = Gτ + TR P ∗ . A finite-difference approximation of the system of Eqs. (9)–(12) is given by δt D =

Here, δt x =

xt+1 −xt−1 2Δt

and xt =

δt ζ = Z, t ∗ ∗ + RGT + TR ln ps ) + D , δt T  = −τ D t + T ∗ , δt ln ps = −πDt + P ∗ .

−∇2 (Φ

xt−1 +xt+1 2

(the sign [ ] is omitted for convenience).

A consequence of this system is the following equation for divergence: )] − Δt2 ∇2 (GT ∗ + TR P ∗ ), (E − BΔt2 ∇2 )Dt = Dt−Δt + Δt[D∗ − ∇2 (Φt−Δt + TR ln pt−Δt s where E is a unit matrix. NUMERICAL ANALYSIS AND APPLICATIONS Vol. 2

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In spectral form, this gives an algebraic equation for each harmonic of D. We have   1 1 t t−Δt t−Δt 2 mt−Δt E + BΔt Dnm = Dnm + Δt[D∗m + TR ln ps nm ] n + Φn n(n + 1) n(n + 1) ∗m +Δt2 (GT ∗m n + TR P n ).

The vorticity, temperature, and surface pressure are found explicitly from (13), (15), and (16), respectively.

2.3. Description of the Experiments The idea behind the numerical experiments whose results are discussed here is to determine and analyze the corresponding variations in the extratropical tropospheric circulation caused by a controlled strengthening of the stratospheric polar vortex [14–15]. The general circulation model was calculated with a spectral resolution T42 on a 128 × 64 grid. In vertical direction 31 levels were treated. In the experiments, the polar vortex strength is determined by the gradient Γ via (8). The following two cases were considered: Γ = 0 and Γ = 4. For the case with zero gradient, a random perturbation of the surface pressure field was given at the initial time. The model was calculated over a period of about two years until reaching a steady state. Additional calculations with a corresponding radiative equilibrium temperature were made for an experiment in which the parameter Γ was equal to 4. The velocity, temperature, and pressure fields averaged over a 15-day period with a step of 1.5 days were taken as diagnostic fields. We analyzed the average pressure, western wind, and temperature as functions of the latitude taken at the lowest level for various values of the radiative equilibrium temperature gradient Γ. This was done to understand how variations of the stratospheric polar vortex affect the atmospheric surface layer. We also constructed an Eliassen–Palm cross section, i.e., a latitude–height crosssection displaying   

vectors of the Eliassen–Palm flux, which in coordinates (y, p) has the form E = −u v  , f vΘθ . (In p

Fig. 6, the Eliassen–Palm flux vectors are shown by arrows, and divergence of the flux, by contours.) The divergence level lines are level lines of the meridional flux of quasigeostrophic potential vorticity. First, u ¯, v¯, and Θ were estimated as quantities averaged over the latitude and time for each of the cases. Then the flux components were calculated for all times. The derivatives with respect to p were approximated by a central difference scheme. Since the computational mode makes a considerable contribution to the distribution of the vertical component in E, averaging over space was used to calculate it. The Eliassen– Palm flux divergence in coordinates (φ, p) has the following form:  = div E

∂(v  θ  /Θp ) 1 ∂(−u v  ) +f . r ∂φ ∂p

The first term is difficult to approximate, for the grid is nonuniform in latitude. Using values at a node and at two neighboring nodes, we obtain the following expression for the derivative with respect to latitude: Ai − Ai−1 φi+1 − φi Ai+1 − Ai φi − φi−1 ∂A = + . ∂φ φi − φi−1 φi+1 − φi−1 φi+1 − φi φi+1 − φi−1 Here, the subscript denotes the node number in latitude. To calculate the second term and the vertical component of the EP flux, the derivative with respect to p was approximated by a central difference scheme. The Eliassen–Palm flux and its divergence are convenient diagnostic tools for examining the influence of vortex motions on the zonal flow. 3. ANALYSIS OF THE RESULTS

3.1. Theory In general circulation models, which can successfully reproduce the angular momentum distribution, stratospheric air can reach extratropical latitudes of troposphere owing to the existence of a mean zonal eddy-induced forcing. This is necessary for the generation of a mean zonal flow through the surfaces of NUMERICAL ANALYSIS AND APPLICATIONS

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constant absolute angular momentum per unit mass. The angular momentum distribution observed in the middle atmosphere is shown in Fig. 1 [6]. The dominant contribution of Earth’s rotation explains the prevalence of the angular momentum vertical distribution in the extratropical zone. For such a distribution of the angular momentum, the extratropical zone is practically inaccessible for the mean circulation without proper forcing. Such forcing exists due to a large-scale angular momentum transfer effected by eddy-induced processes, such as the breakdown of Rossby waves (or gravitational waves) propagating upward from the troposphere. A linear theory of vertically propagating Rossby waves proposed in [6] still serves as a good basis for gaining insight into the dynamics of troposphere–stratosphere interactions. The eddy-induced processes can be considered, in a sense, as effects determining (controlling) the intensity and distribution of the mean circulation in the extratropical zone on long-period time scales. Such a control does not ignore other factors (for instance, a feedback between the mean atmospheric state and the generation and dissipation of upward propagating Rossby (gravitational) waves. It is important that for this angular momentum distribution, the influence of eddy-induced forcing on the mean extratropical circulation over a rather long period of time compared to the radiative relaxation time is directed toward the troposphere. For instance, if the forcing is concentrated in some layer, the streamlines of the mean circulation will be directed downward and will vanish above this layer. The rate of stratospheric air particle penetration into the extratropical troposphere, for instance, through an isentropic surface, is controlled by the dissipation of eddies at the overlying surface. A research question of interest is to determine the degree of influence of stratospheric dynamics on the surface climate and its variability. The available observations show that the amplitudes of major modes of atmospheric variability [8–14] in the entire atmosphere—from the surface to the stratosphere—have a tendency that is closely related to increasing temperature at high latitudes in the northern hemisphere. However, there is no clear understanding of the details of how the stratosphere affects the tropospheric dynamics in the northern hemisphere. A principal purpose of investigating the atmospheric dynamics is to understand the dynamics of low-frequency variability of the extratropical atmosphere. This type of variability is characterized by time scales of 10 or more days and by a rather simple spatial wave structure, which makes a major contribution to the variability. The available observational data also indicate that the extratropical troposphere–stratosphere coupling variability manifests itself mainly in the zonally symmetric component of a flow [8–11]. This phenomenon is known as Arctic Oscillations/North-Atlantic Oscillations (AOs/NAOs), or annular modes of atmospheric oscillations in the northern hemisphere. Proper observations show that the total amount of ozone in air column above the Arctic correlates negatively with AOs. It is, however, assumed that the AO index increases with strengthening tropospheric polar jet, which is usually accompanied by a decrease in the tropopause and, hence, by

Fig. 1. Distribution of mean zonal absolute angular momentum in the middle atmosphere for January and July (units: 6.4 · 107 m2 /s). NUMERICAL ANALYSIS AND APPLICATIONS Vol. 2

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Fig. 2. Relation between NAO modulations, tropopause height, and stratospheric vortex intensity.

an increase in the stratospheric air column. A resolution of this paradox is mostly connected with an NAO stratospheric component. It has been found that the NAO index and the intensity of the stratospheric polar vortex are closely related and strongly affect the tropopause and sea level pressure [14]. This dependence is shown schematically in Fig. 2. When the NAO index increases due to a cyclonic anomaly over Iceland (circular arrow over IC) and the tropopause (solid line) decreases, the Rossby waves propagating upward (wavy lines) turn toward the equator and interact less with a jet flow. The jet flow expands, and there appears a positive anomaly of potential vorticity (+). The tropopause below this anomaly rises, and the air column stretching (vertical arrows) enhances the cyclonic circulation above the north pole (circular arrow over NP). Recall that the NAO index is defined as the projection onto the first leading mode of natural orthogonal function (NOF) of the sea-level pressure within the Atlantic region. A seasonal cycle is removed from each time series of the projections thus obtained. The Deep Icelandic Low, which has a low tropopause and is related to a positive NAO, is also caused by Rossby waves propagating upward and turning toward the equator. Hence, the breakdown of waves in the vicinity of a stratospheric jet decreases (the jet deceleration weakens). This enhances the jet, generates a positive anomaly of potential vorticity over the pole, stretches the vortex tube into the troposphere, and rises the tropopause. A relative vorticity generation over the pole is accompanied by a decrease in pressure near the pole. Analysis of the equilibrium state of the atmosphere obtained by numerical modeling shows that when the polar stratosphere is cooling, the tropospheric jet axis turns toward the pole, and the surface pressure in the polar zone drops. In this case the vertical flux of wave activity from the troposphere to the stratosphere decreases, whereas the meridional flux of wave activity from high latitudes to midlatitudes increases. This means that when the polar stratosphere is cooling, the wave impedance in the stratosphere drops. The structure of such a reaction is similar to that of leading modes of the inner variability of the atmosphere in extratropical latitudes of the northern hemisphere, which are called North-Atlantic Oscillations (NAOs); often, this is a manifestation of the Northern Annular Mode (NAM) [8, 9, 13]. The annular mode, defined as the leading natural orthogonal function (NOF) of the mean zonal surface pressure, has a dipole meridional structure. It is well known that the NOF of a zonally varying pressure has the same meridional structure and is a zonally symmetric mode. It should be observed, however, that the annular mode must be interpreted as a zonally uniform distribution of zonally localized circulations with the same meridional structure rather than as a zonally symmetric oscillation mode. Often the annular mode manifests itself as a series of waves of a meridional structure similar to that of a symmetric mode with a clear zonal structure extending from 60 to 90 degrees of longitude with a small inclination in the direction of north-west and south-east between the centers of action. North-Atlantic Oscillations are the most prominent wave structure in the northern hemisphere [16]. The NAOs affect temperature and precipitation over a large region, from Europe to Northern America, which is of great interest for long-range weather predictions. It is also important that the NAM has an equivalent barotropic vertical structure that extends up to the stratosphere and higher. NUMERICAL ANALYSIS AND APPLICATIONS

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Fig. 3. Latitude-longitude distributions of temperature (K) for the northern hemisphere in coordinates (φ, σ), where φ is latitude, σ is determined by relation (1); left figure: Γ = 0, right figure: Γ = 4.

Fig. 4. Latitude-longitude distributions of zonal velocity (m/s) for the northern hemisphere in coordinates (φ, σ). Dark color: zones of eastern wind; left figure: Γ = 0, right figure: Γ = 4.

Figures 3 and 4 reproduce distributions of temperature and zonal velocity for a weak or strong polar vortex. The mean zonal wind, pressure, and surface temperature versus latitude for both experiments are shown in Fig. 5. For a zero gradient, the zonal velocity field is symmetric about the equator. Looking at Fig. 5, we see that western winds with velocities of 10 m/s in the troposphere are dominant at mid-latitudes. Eastern NUMERICAL ANALYSIS AND APPLICATIONS Vol. 2

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Fig. 5. Dynamic fields at the surface versus latitude: (a) mean zonal wind (m/s); (b) pressure (mbar); (c) surface temperature (K). Latitude range: from 0◦ to 90◦ . In all figures, results for Γ = 0 are shown by the solid line, and for Γ = 4, by the dashed line.

winds of 10 m/s prevail in the equator zone. In the stratosphere, the maximum western wind velocity is equal to 25 m/s at zero gradient and to 50 m/s at maximum polar vortex. As the polar vortex strengthens in the stratosphere, the pressure in the lower polar troposphere decreases and, in accordance with quasigeostrophic motion formulas, the western wind in the polar region strengthens. At the same time, we can observe some warming in the troposphere, substantial in the polar cap zone and small outside this zone.

3.2. Eliassen–Palm Flux A peculiarity of the dynamics of troposphere–stratosphere interactions is that, as a rule, the waves generated in the troposphere propagate upward, whereas the mean zonal flow variations propagate downward. The downward propagation of anomalies is triggered by different mechanisms. A balanced response to wave impedance is a change in the zonal flow and generation of a meridional circulation. The meridional circulation generation causes a surface pressure change. A rotational moment induced by wave impedance makes a partial contribution to the change in relative angular momentum and in global angular momentum (through the distribution of mass). Therefore, to investigate the dynamics of interaction, we use standard diagnosis based on the theory of Eliassen–Palm fluxes, which is most appropriate in this setting. The concept of potential vorticity (PV), q, is conveniently used in quasigeostrophic theory to diagnose processes taking place in atmosphere. Indeed, under conditions of adiabatic motion, this quantity is a passive tracer, which determines the atmospheric dynamics uniquely. Therefore, we can make some conjectures as to the character of air mass motion and its properties, such as humidity and pollutant concentrations. The meridional vorticity flux Q = v  q  is an indicator of vorticity transfer. Under the conventional hydrostatic assumptions, the amount of PV per unit volume is equal to the ∂v − ∂u absolute “isentropic” vorticity, which can be written in the form ζaθ = ζθ + f , where ζθ = ∂x ∂y , f is the Coriolis parameter, and derivatives are taken at a constant potential temperature. In the midstratosphere, isentropic surfaces almost coincide with isobaric surfaces. Therefore, potential vorticity maps nearly coincide with absolute vorticity maps, and absolute vorticity fluxes, as well as PV fluxes, can help us gain insight into the character of interactions between various atmospheric layers. Meridional fluxes are positive in the mid-latitudes (from 30◦ to 65◦ ) and are negative in the tropics. That is, at midlatitudes, air masses with high vorticity are transported to the poles. A characteristic feature of a field with large temperature gradient is the presence of a zone of intensive vorticity concentration in the upper stratosphere. With enlarged vorticity gradient, the zone of vortex transport to the poles narrows abruptly. To understand the dynamics of the mid-latitudes, we should take into account feedbacks between the momentum and heat vorticity fluxes and the main flux. Therefore, the model scope has to be NUMERICAL ANALYSIS AND APPLICATIONS

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expanded. With allowance for the above processes, the mean zone circulation in the quasigeostrophic approximation is described by the following system of equations in the (y, z) plane on the sphere: ∇•E ∂u ¯ − f · v¯∗ = (v  q  ) = , ∂t aρ0 cos φ

(17)

∂ θ¯ ∂ θ¯ ¯ +w ¯∗ = Q, ∂t ∂z

(18)

∂¯ v∗ ∂w ¯∗ + = 0, ∂z a∂φ

(19)

where (v  q  ) is the mean flux of quasigeostrophic potential vorticity, ∇ · E is the Eliassen–Palm flux  −1 ¯ is a heat source, Ez = f ρ0 a cos φ(v  θ  ) ∂ θ¯ divergence, E = (Eφ , Ez ), Eφ = −ρ0 a cos φ(v  u ), Q , ∂z   −1 ¯ ∂ (v  θ  ) ∂∂zθ is the modified mean zonal meridional velocity of mean circulation, w ¯∗ = v¯∗ = v¯ − ∂z   −1 ¯ ∂ is the modified (nonadiabatic) mean zonal vertical velocity of mean circulation. w ¯ + a∂φ (v  θ  ) ∂∂zθ In view of (17)–(19), the modified vertical velocity of mean circulation and the mean wave forcing in the layer above the level being considered are related via ⎧∞ ⎫  ⎨ ⎬ 1 ∂ ∇•E dη . (20) w ¯∗ = − 2 a ρ0 cos φ ∂φ ⎩ 2Ω sin φ ⎭ z

Relation (20) is a well-known downward control principle [6] in the quasigeostrophic approximation and under conditions of steady-state mean circulation. Hence, it is the convergence of total wave energy flux that regulates an energy exchange between the waves and the zonal flow. The direction of the Eliassen– Palm flux vector is an indicator of relative contributions of heat and momentum fluxes in their total contribution to the potential vorticity transfer. It can be shown that whenever the concept of group velocity c is applicable, we have E = cA, where A = a quantitative measure of wave propagation.

1 q 2 2 q¯y

is the wave activity. Thus, E, in a sense, is

The Eliassen–Palm  flux vector determined in the (y, pt) plane and written in dimensionless form  v θ   was used as a diagnostic quantity. Analysis of Fig. 6 shows that the EP flux as E = −u v , f0 Θ p

below goes upward from the sources near the surface, and then turns toward the equator in the upper troposphere. From the distribution of the EP flux horizontal component difference we can see that the momentum transfer with a nonuniform distribution of the background temperature in high latitudes is greater than is with a radiative equilibrium temperature independent of the latitude. Vast tropospheric zones, in which the horizontal component is directed toward the equator, are observed in both cases. Yet, at zero gradient, these zones are concentrated in the vicinity of 40◦ , and at nonzero gradient, the zone of intensive horizontal transfer is located closer to the pole. At mid-latitudes, the vertical component of the EP flux is much greater in the case of a large gradient. The Eliassen–Palm flux divergence at mid-latitudes is positive in the surface layer and is negative in the upper troposphere. Thus, we may conclude that, in our model, strengthening polar vortex increases the vorticity transfer and energy exchange between different atmospheric layers. Note that upward transfer of the wave activity of quasigeostrophic waves means that the vortex heat flux (f [v  θ  ] > 0) is directed toward the pole regardless of the mid-latitude temperature gradient, and the wave activity transfer to the equator indicates that the vortex flux of momentum ([u v  ] > 0) is directed toward the pole. NUMERICAL ANALYSIS AND APPLICATIONS Vol. 2

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Fig. 6. Eliassen–Palm cross-section (in coordinates (latitude, σ), where σ is determined by relation (1), only the northern hemisphere is considered): Eliassen–Palm (EP) flux vectors and divergence level lines (m/s2 ) (a) Γ = 0; (b) Γ = 4; (c) difference. NUMERICAL ANALYSIS AND APPLICATIONS

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4. CONCLUSIONS A spectral model of atmospheric general circulation with a simple zonally symmetric heat source of Newtonian form, Rayleigh friction at vertical boundaries, and homogeneous underlying surface (no orography and thermal contrasts of the underlying surface) was used to investigate the reaction of the stratosphere-atmosphere system to variations in the stratospheric thermal structure. Although numerical experiments were performed with a relatively simple model of atmospheric general circulation, it has been shown that disturbances in the thermal stratification of the stratosphere can cause the following changes in the circulation in the troposphere and at the surface: • the zonal flow jet in the lower troposphere turns toward the pole and the stratospheric polar vortex strengthens as Γ increases; • the surface pressure in the polar zone drops as Γ increases; • the vertical flux of wave activity from the troposphere to the stratosphere decreases with strengthening stratospheric vortex, since it turns toward the equator, which decreases wave impedance in the lower stratosphere; • the dynamic lower troposphere-lower stratosphere interaction was investigated with Eliassen– Palm fluxes whose divergence has a direct bearing on the circulation in the lower troposphere; therefore, the above analysis cannot be considered complete; • experimental results show that the response of the troposphere to strengthening stratospheric polar vortex correlates well with a positive NAO index. This indirectly confirms the hypothesis that the first leading NAO mode, which has stratospheric and tropospheric components, exerts a considerable impact on the interaction process. The experiment with the general circulation model has shown that the sensitivity of the troposphere to stratospheric cooling is partly related to interactions between the mean zonal flow and synoptic- and planetary-scale vortices. ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research, project no. 05-05-64989. REFERENCES 1. Hoskins, B.J. and Simmons, A., A Multi-Layer Spectral Model and the Semi-Implicit Method, Q.J.R. Met. Soc., 1975, vol. 112, pp. 1231–1250. 2. Held, I.M. and Suarez, M., A Proposal for the Intercomparison of the Dynamical Cores of Atmospheric General Circulation Model, Bull. Am. Meteor. Soc., 1994, vol. 75, pp. 1825–1830. 3. Krupchatnikoff, V.N. and Kurbatkin, G.P., Modelirovanie krupnomasshtabnoi dinamiki atmosfery. Chislennye metody (Modeling of Large-Scale Atmospheric Dynamics. Numerical Methods), Novosibirsk: Comp. Center, Sib. Branch of USSR Acad. Sci., 1991. 4. Krupchatnikoff, V.N. and Kurbatkin, G.P., Modelirovanie krupnomasshtabnoi dinamiki atmosfery. Metody diagnoza obshchei tsirkulyatsii (Modeling of Large-Scale Atmospheric Dynamics. Diagnostic Methods of General Circulation), Novosibirsk: Comp. Center, Sib. Branch of USSR Acad. Sci., 1991. 5. Charney, J.G. and Drazin, P.G., Propagation of Planetary-Scale Disturbances from the Lower into the Upper Atmosphere, J. Geophys. Res., 1961, vol. 66, pp. 83–109. 6. Haynes, P., Marks, C., McIntyre, M., Sheperd, T., and Shine, K., On the “Downward Control” of Extratropical Circulation by Eddy-Induced Mean Zonal Forces, J. Atm. Sci., 1991, vol. 48, pp. 651–678. 7. Haynes, P., Stratospheric Dynamics, Ann. Rev. Fluid Mech., 2005, vol. 37, pp. 263–293. 8. Volodin, E.M. and Galin, V.Ya., Investigation of the First Mode of Low-Frequency Variability of Atmospheric Circulation in Mid Latitudes of the Northern Hemisphere, Meteor. Gidrol., 1998, no. 9, pp. 26–40. 9. Thompson, D.W. and Wallace, J.M., The Arctic Oscillation Signature in the Wintertime Geopotential Height and Temperature Fields, Geophys. Res. Lett., 1998, vol. 25, pp. 1297–1300. NUMERICAL ANALYSIS AND APPLICATIONS Vol. 2

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