PHY2504 Course Project: Zonal Momentum Balance, Entropy Transport, and the Tropopause

Transcription

1 PHY2504 Course Project: Zonal Momentum Balance, Entropy Transport, and the Tropopause Andre R. Erler April 21 th,

2 Contents 1 Introduction 3 2 Zonal Momentum Balance in Isentropic Mass Flux Zonally Averaged Momentum Balance and Boundary Conditions Eddy Fluxes and the Mean Isentropic Circulation Verification of the Predictions Discussion Comparison with Quasi Geostrophic Theory The Thermal Stratification of the Troposphere and other Applications The Vertical Extend of Eddy Mixing An Estimate for the Height of the Tropopause Summary and Conclusion 30 2

3 1 Introduction A major focus of atmospheric dynamics is to predict the general circulation of planetary atmospheres as a function of external parameters such as the rotation rate, the equator to pole temperature contrast, atmospheric composition, and the lower boundary condition. General Circulation Models (GCMs) are now able to simulate a wide range of climates under different forcings, but they study of GCM results gives little theoretical insight. For instance we come to believe that the static stability of the extra tropical troposphere is determined by baroclinic eddies; only we are still not able to predict the observed value from theoretical considerations (based on external parameters). Here I will summarize and discuss an attempt to capture the qualitative features of the tropospheric circulation, due to Schneider (2004, 2005, 2006). The approach is based on the idea that the tropospheric circulation is driven by the redistribution of entropy poleward and upward. The mathematical framework is based on consideration of momentum and mass fluxes on isentropes. Schneider (2005) focuses on the role of thermal variables (entropy and stratification), rather than dynamical (momentum, relative vorticity); quasi geostrophic theory will be inadequate for this purpose since it can only predict small perturbations of the former around a reference profile. 1 The General Circulation along Isentropes Arguably the approach of Schneider (2005) was motivated by the much simpler form of the hemispherical overturning circulation in isentropic coordinates. Fig. 1 displays the stream function of the meridional overturning circulation in pressure 1 This limitation of QG theory may partly explain the general preoccupation with dynamical structures; it may even have lead to an overestimation of the relevance of relative vorticity (and wind) in the troposphere; also cf. Sun and Lindzen (1994). 3

4 coordinates (top); there are three major overturning cells: the Hadley cell, which is dominated by a direct thermal circulation and diabatic heating at the equator, the eddy driven Ferrell cell in mid latitudes, and the polar cell, dominated by radiative cooling. The corresponding stream function in isentropic coordinates is displayed in Fig. 2 (top): it exhibits a much simpler structure with only a single overturning cell. In the (Fig. 1, top) it is difficult to see how heat (or entropy) is transported from the equator to the pole; with the isentropic stream function (Fig. 2, top) it is obvious: entropy/heat is received at the equator and redistributed poleward over the mid latitudes (as streamlines descend gradually towards the pole). Fig. 1 (bottom) shows the climatological pressure height of isentropes; note that isentropes in the mid latitudes slope upward and poleward, and intersect the ground. The isentropic heating rate (cross isentropic flow) is displayed in Fig. 2 (bottom): in the interior atmosphere on the poleward branch diabatic cooling is slow and relatively homogeneous, but in the surface layer on the equatorward branch cross isentropic flow is concentrated at the surface. This is related to the fact that isentrope intersect the ground, and a theoretical description of this phenomenon will be a major focus of the next section. Also note that the heating displayed in Fig. 2 (bottom) is an average value; in fact the heating occurs in a very localized manner, so that cold isentropes can extend very far towards the equator before they heat up. Surface potential temperature fluctuations are large. 2 Zonal Momentum Balance in Isentropic Mass Flux In this section I will sketch the derivation of and discuss an equation that relates isentropic mass flux to eddy fluxes of potential vorticity and surface potential temperature. The derivation in its complete form (and most of the material in this section) was originally 4

5 Figure 1: Top panel: the mean meridional stream function. Three distinct overturning cells are visible: the tropical Hadley cell, the eddy driven Ferrell cell in mid latitudes, and the polar cell. Compare this to Fig. 2 (top). Bottom panel: annual mean isentropes in pressure coordinates. Note the slope of the isentropes in the extra tropical troposphere. The thermal stratification can be estimated from the isentrope spacing: it is high in the stratosphere and lower in the troposphere.(kallenberg et al., 2005) 5

6 Figure 2: Top panel: the annually averaged mass flux stream function in isentropic coordinates. The isentropic overturning cells span the entire hemispheres, but the Hadley cell is still discernable as a local extremum. The mass flux in the upper branch is poleward, in the lower branch it is equatorward. A significant part of the equatorward circulation branch occurs on isentropes which frequently intersect the surface. Bottom panel: significant cross isentropic flow occurs on near surface isentropes, while diabatic heating/cooling in the interior atmosphere is small. (Kallenberg et al., 2005) 6

7 given by Schneider (2005). 2.1 Zonally Averaged Momentum Balance and Boundary Conditions The point of departure is mass and momentum conservation in isentropic coordinates. For the purpose of this derivation we will assume an idealized statistically stationary and axisymmetric atmosphere. The surface will be assumed to be flat, and we will only consider dry adiabatic dynamics (of an ideal gas). As external parameters we only require external forcing, i.e. diabatic heating and friction, and the Coriolis parameter. Note that the static stability will be determined internally, and should ideally be predicted by the theory. In isentropic coordinate adiabatic motion is two dimensional (i.e. confined to isentropes); the equivalent of density in isentropic coordinates is σ = 1 g θp, which is essentially the mass ( g p) between isentropic layers. Isentropic mass conservation in flux form then takes the form t σ + x (σu) + y (σv) + θ (σq) = 0, (1) with Q = θ denoting the diabatic cross isentropic velocity and the horizontal derivatives understood to be taken along isentropic surfaces. Similarly momentum conservation can be written in isentropic form with uσ as the zonal component of the momentum density; again suppressing all metric factors due to curvature of the sphere and sloping of isentropes we can write t (σu) f σv + x (σu 2 ) + y (σuv) + θ (σuq) = σ x M + σf x. (2) This form of the momentum equation is formally analogous to the primitive equations in log p coordinates, with the Montgomery potential M = Φ + c p T assuming the role of 7

8 the geopotential Φ; the Montgomery potential is essentially the dry static energy, i.e. the potential energy plus the heat energy. If, for a moment, we ignore the diabatic cross isentropic flux, the equations of motion are formally similar to the shallow water equations: the isentropic density assumes the role of the layer thickness and the Montgomery stream function acts like a bottom topography. The main difference to shallow water models is that isentropes can intersect the ground at varying locations, so that the lower boundary condition requires some more care. Surface potential temperature θ s is known to change dynamically, in the real atmosphere and also in continuous quasi geostrophic models. An isentrope of a potential temperature lower than the instantaneous surface potential temperature lies beneath the surface. Isentropes beneath the surface (θ < θ s ) do not contain any mass, so it is natural to treat the isentropic density beneath the surface as zero. To formalize this convention we write σ σh(θ θ s ), θ. (3) Equations (1) and (2) can now be combined; using definition (3) and with ζ θ = y u x v denoting the relative vorticity perpendicular to isentropic surfaces, we arrive at [ t u (f + ζ θ ) v] σh(θ θ s ) = [ x B Q θ u + F x ] σh(θ θ s ). (4) B = ((v 2 + u 2 )/2 + gz + c p T ) is the Bernoulli function, which essentially represents the total energy of the fluid at a given location (in stationary flow B is constant along streamlines). Note that both sides of the equality (4) are multiplied by the term σh(θ θ s ), so that σ cancels in the interior atmosphere (θ i > θ s t), where isentropes do not intersect the ground (and σ > 0 θ i ). t u (f + ζ θ ) v = x B Q θ u + F x θ > θ s (5) 8

9 At this point assume stationary and axisymmetric circulation statistics, and average over the zonal direction. The common zonal average is defined as φ(x) = 1 x x2 x 1 φ(x) dx, and the mass weighted average defined as φ(x) = σ(x)φ(x)/ σ. In a periodic domain zonal derivatives vanish by definition x φ(x) = [φ(x)] x 2 x 1 zonal and temporal averaging reduces equation (5) to = 0, so that in the interior atmosphere vp + J Q + JF = 0. (6) P = (f + ζ θ ) /σ is the potential vorticity on isentropes, and J Q = Q θ u/σ and J F = F x /σ are the diabatic terms. Note that P, J Q, and J F are as of now only defined in the interior atmosphere. On isentropes which intersect the surface, values have to be assigned to all quantities beneath the surface. The following definitions are essentially based on conventions, but they are physically motivated by the idea, that isentropes which lie below the instantaneous surface potential temperature are not actually inside the ground but are infinitely thin layers (of vanishing mass) tracing the surface. Pressure p = p s, θ < θ s, is set to the instantaneous surface pressure Velocity (u, v) = 0, θ < θ s vanishes, in accord with the no slip boundary condition Relative vorticity ζ θ = 0, θ < θ s also vanishes (Convention I in Schneider, 2005) 2 2 An alternative convention also discussed by Schneider (2005) sets the absolute vorticity σp = f +ζ θ = 0 in subsurface isentropes to zero (Convention II in Schneider, 2005). We will not use this convention here as it proved to be of no relevance for later work and also implies that the relative vorticity ζ θ = f attains its largest magnitude inside the surface, which I regard as rather unphysical and it violates continuity with the no slip boundary conditions. Convention II may have arisen from the desire to define a finite potential vorticity inside the surface: Convention I implies P = f/σ = infinite PV if σ = 0 vanishes. However, the layer is massless, so that a meaningfull finite value can be assigned in a mass weighted average: P = σσp/σ = σ f θ < θ s. 9

10 The diabatic terms are also set to zero: J Q J Q H(θ θ s ) and J F J F H(θ θ s ) Note that the mean of the step function can assume a non integer value 0 H(θ θ s ) 1. With the above definitions the average over surface layer isentropes can be evaluated. A complication we have to consider is that the surface potential temperature can vary in a zonally asymmetric way, so that isentropes can attach to and detach from the surface along a small circle of given latitude. In a statistically stationary state with the no slip boundary condition the acceleration term vanishes along a small circle. The term involving the Bernoulli function is determined by the boundary contributions (the step function vanishes beneath the surface). The wind contribution [u 2 + v 2 ] x 2 x 1 = 0 vanishes immediately due to the no slip boundary condition. The Montgomery stream function can assume different values at the boundary points; it can be treated by integration by parts: x B H(θ θ s ) = 1 x2 x x 1 B(x)H (θ θ s (x)) dx = 1 x [BH(θ θ s)] x 2 x 1 1 x x2 x 1 B x H(θ θ s ) dx = Mδ(θ θ s ) x θ s. (7) The step function H(θ θ s ) vanishes at the boundary (x 1, x 2 ), so only the integral term survives (which is equivalent to a zonal average). With x H(θ θ s (x)) = δ(θ θ s ) x θ s, equation (7) follows. The form drag term arises from (Montgomery) potential differences between the points where isentropes attach and detach from the surface. A similar term also arises in log p and pressure coordinates from the intersection of pressure surfaces with the topography, either due to surface pressure change or uneven topography. The zonally averaged zonal momentum balance in isentropic coordinates valid in the 10

11 interior as well as the surface layer now reads σvp + σj Q + σjf = Mδ(θ θs ) x θ s. (8) We now decompose the meridional PV flux into mean flow and eddy components vp = v P + v P and write σv = 1 P ( σv P + σj Q + σjf ) + 1 P Mδ(θ θ s ) x θ s ; (9) since we divided by P, we are now restricted to regions where P does not vanish, i.e. away from the equator. Note that, because we are using mass weighted averages, the term σv = σ σv/ σ = σv in fact represents the total meridional mass flux, including eddy fluxes. Furthermore, because entropy is materially conserved and constant along isentropes, σv can also be interpreted as an entropy flux. The form drag term will assume a more readily interpreted form after vertical integration; ( for this purpose we will use the approximation P (θ) ) 1 ( θs P (θ s ) ) 1 and introduce the balanced surface eddy velocity ṽ s = f 1 x M s: 3 θint θ min M x θ s δ(θ θ s ) P dθ 1 P ( θ s ) θint x M θ s δ(θ θ s ) dθ f θ min P ( θ s ) ṽ sθ s s. (10) The lower boundary of integration θ min is the lowest potential temperature that occurs at the surface and the upper boundary θ int is some potential temperature in the interior atmosphere (i.e. an isentrope that never touches the ground). The vertically integrated isentropic mass flux (with the same boundaries) then takes the form θint θ min θint σv σv dθ P + σjq + σjf θ min P dθ f P ( θ s ) ṽ sθ s s. (11) 3 Schneider (2005) gives a slightly different definition: ṽ s = f 1 x (M s c p θ s), presumably to draw an analogy to quasi geostrophic theory. The modification is of no consequence since the term x θ s drops out after averaging: θ s x θ s = x (θ sθ s) = 0. 11

12 Or, after neglecting diabatic heating J Q and using the planetary vorticity approximation P f/ σ (valid at small Rossby number), a simpler form can be written as θint θint σv σv dθ P + σjf θ min P dθ σ( θ s )ṽ sθ s s. (12) θ min The simplified form (12) is the point of departure for Schneider (2004). Equations (11) and (12) essentially relate the isentropic mass flux (mean and eddy contributions) to eddy fluxes of potential vorticity and surface potential temperature plus an Ekman term due to friction and diabatic (cross isentropic) contributions. The eddy flux terms will be the most important contributions and are dynamically the most interesting. The only approximations made so far are related to small Rossby number, hydrostasy, and the no slip boundary condition. We also did not consider variable topography and assumed a quasi stationary state. Moist processes and radiative forcing are retained in (11) in form of diabatic terms, but were neglected in (12). We also assumed an ideal gas atmosphere, but for earth that is almost exact. If an universally applicable eddy flux closure for isentropic PV and surface potential temperature was available, (12) would in principle allow the inference of the mean isentropic circulation, and with it the efficiency of entropy (heat) redistribution in the atmosphere. Unfortunately such an universal eddy flux closure is, as of now, not available an we have to resort to very rough approximations, only valid on planetary scales. 2.2 Eddy Fluxes and the Mean Isentropic Circulation From this point the strategy will be to find physically reasonable approximations to the eddy flux terms, which will allow us to make some inferences about the mean meridional mass and heat flux on the basis of (12). The arguments will be of a very general nature and can in principle be applied to any planetary atmosphere with a rigid lower boundary (with no slip condition), low Rossby 12

13 number (fast rotation), and stable stratification with respect to adiabatic vertical overturning. 4 In order to estimate PV gradients we will again assume that the planetary contribution F dominates the vorticity budget, so that the potential vorticity is determined by the thermal stratification P f σ. (13) On average this approximation holds very well. Furthermore we will be interested in the meridional PV gradient on isentropes, given by ( ) y P = y = f σ β σ f y σ. (14) σ 2 The Interior Atmosphere As before, the interior atmosphere is somewhat easier to treat. Isentropes in the interior do not touch the surface, and are on average well removed from the boundary layer, so we can neglect Ekman fluxes. Thus (12) reduces to θint θ min θint σv σv dθ P θ min P dθ. (15) Expecting some kind of diffusive behaviour, such that eddy fluxes are directed down gradient, we assume v P to be proportional to the gradient of PV along isentropes (14). We can write θint where D i θ min θint σv dθ θ min σ y P θint ( P dθ D i y σ σ β ) dθ, (16) θ min f is an eddy diffusivity. In the earth atmosphere the β term in (16) is small compared to the gradient in isentropic density. The reason for this is that isentropes slope upward towards the pole and intersect the tropopause; the major contribution comes from 4 In convectively dominated atmospheres (e.g. those of gas giants) isentropic coordinates loose their applicability. 13

14 the strong increase in static stability (decrease in isentropic density) at the tropopause (a factor of four, Schneider, 2004). We conclude that the mass flux on isentropes intersecting the tropopause (and not touching the surface) will be directed poleward. In the earth atmosphere this is true for most of the interior mid latitude atmosphere (the so called middle world). The Near Surface Layer To estimate the isentropic mass flux in the surface layer we will again assume that eddy fluxes are directed down gradient. The direction of the surface potential temperature gradient y θ s is obviously directed equatorward, and this holds for most planetary atmospheres. With the PV gradient (14), we can write θint θint ) σv βf σ2 dθ (D i y σ D i σ + f J F dθ + σ( θ s ) D s y θs, (17) θ min θ min where D i and D s are eddy diffusivities in the surface layer and at the surface, respectively. In order to infere the direction of isentropic mass fluxes we have to estimate the average gradient of isentropic density on isentropes intersecting the surface. From definition (3) we expect the average isentropic density on a near surface layer to be proportional to the frequency of the respective isentrope being above the surface; we can write σ Π(θ s ) σ( θ s ), where Π(θ s ) is the cumulative distribution function of surface potential temperature at a given latitude. β term is directed poleward, and Ekman fluxes also in regions of surface westerlies, however, they are of higher order in σ, which is small near the surface, so that there will be a layer where the PV gradient term dominates. We conclude that the poleward gradient of isentropic density (equatorward gradient of PV) and the equatorward gradient of surface potential temperature dominate over the other terms and imply an equatorward mass flux on surface layer isentropes. 5 5 Infact both estimates, the PV gradient and the gradient of θ s, depend on the distribution Π(θ s ) of 14

15 2.2.1 Verification of the Predictions Schneider (2005) conducted highly idealized GCM experiments and computed the isentropic mass flux, corresponding to the theory developed above. The GCM Schneider (2005) used consisted only of a dynamical core, without any sophisticated representation of other physical processes in the atmosphere. The Surface was flat with a prescribed skin temperature and the dynamics were forced by a relaxation temperature profile. It is noteworthy that convective adjustment was only used in the tropics (to simulate the Hadley cell). In the extra tropics the relaxation profile is convectively unstable in the lower troposphere (corresponding to radiative equilibrium alone), and the resulting stable stratification is maintained by baroclinic eddy activity. Fig. 3 shows the gradient of the mass weighted average potential vorticity (left) and the isentropic mass flux stream function (right) from a GCM experiment with earth like parameters and boundary conditions. The PV gradient was computed according to the convention adopted here (Convention I in Schneider, 2005). The qualitative agreement of the estimates given above with the GCM experiments is very good: equatorward isentropic mass transport occurs mainly on isentropes which frequently intersect the surface, while the mass transport in the interior atmosphere is poleward. The PV gradient in the surface layer is opposite to the interior atmosphere, also in agreement with the prediction. The estimate of the isentropic mass flux also compares favourably with the observations displayed in Fig. 4 (left panel, also cf. Fig. 2, top): the mass flux in the interior atmosphere is directed poleward, while the surface layer flux is directed equatorward. Some streamlines in Fig. 4 (left) close within the tropics and the mid latitudes; this was not predicted by the theory, but it is associated with diabatic effects and the Hadley cell which we did not consider here. Fig. 4 (right) shows the mass flux stream function computed with respect surface potential temperature with latitude. 15

16 Figure 3: The mean PV gradient (left) and the mean isentropic mass flux stream function (right) in isentropic coordinates obtained from an idealized GCM integration. The dotted line indicates the median surface potential temperature. The PV gradient is negative in the surface layer which intersects the ground, and positive in the interior atmosphere (as in the Two layer QG model by Philips). Similarly the surface layer contains most of the equatorward branch of the isentropic mass flux, while the mass flux in the interior atmosphere is poleward. Also indicated on the right panel is the tropopause height (thick solid line). (Schneider, 2004, 2005) 16

17 Figure 4: Annual mean isentropic mas flux stream functions in isentropic coordinates. The left panel shows the conventional dry isentropic transport, the right panel moist isentropic mass flux. In moist isentropic coordinates the mass flux appears much more regular in the meridional direction and the Hadley cell disappears. Also indicated are the median surface potential temperature (solid black line) and the 10 th and 90 th percentile (dotted lines). (Pauluis et al., 2008, NCEP NCAR Reanalysis data) 17

18 to moist isentropes: the Hadley cell disappears and the stream function looks relatively symmetric about the mid latitudes (Pauluis et al., 2008). 2.3 Discussion The theory developed by Schneider (2004, 2005) aims to represents the global circulation in terms of hemisphere spanning entropy transport cells. The idea is that the atmospheric circulation is driven by the redistribution of entropy received at the surface in the tropics to the top of the atmosphere in high latitudes (where it is radiated to space). This goal is partly achieved by the choice of isentropic coordinates. The moist isentropic stream function suggests that the theory can be extended to moist isentropes, so that latent heat flux can be included in the isentropic overturning cell. However the theory cannot predict the isentropic mass flux stream function from external parameters. Nevertheless we gained insight into the mechanisms of isentropic transport and we are able to explain qualitative features of the overturning cell. In my view the insights we gained are of a very general nature in that they apply to a wide range of planetary atmospheres: the derivation of (11) only requires the lower boundary condition, fast rotation (small Rossby number), and stable stratification, so that we can use the planetary vorticity approximation and isentropic coordinates. If we accept the diffusive Ansatz for eddy fluxes, we only need very few external parameters to estimate general feature of the isentropic overturning circulation. In order to arrive at the estimate (17) for the surface layer potential vorticity gradient and the surface potential temperature gradient we only assumed that a temperature gradient exists, pointing from the pole to the equator; this will almost always be true for terrestrialplanets as long as the axis of rotation is roughly perpendicular to the ecliptic plane. To estimate the potential vorticity gradient in the interior atmosphere (16) we assumed the existence of a tropopause and that isentropes intersect the tropopause in high latitudes 18

19 towards the pole. This may appear arbitrary but is in fact also very generic for terrestrialtype planetary atmospheres. The assumption is essentially equivalent to assuming that the isentropic mass flux closes within the troposphere and that the overturning circulation deposits entropy/heat along its way to the pole (i.e. that streamlines in isentropic coordinates descend towards the pole). The former is almost by definition (as we will see in the next section), and the latter is again a consequence of the pole to equator temperature gradient. Finally we have to assume that the static stability in the stratosphere is significantly higher than in the troposphere; this is supported by radiative transfer calculations. However the last assumption does depend on the chemical composition and distribution of aerosol in the atmosphere and can in principle render our conclusions invalid. As long as the optical thickness is proportional to a near hydrostatic density profile, the assumption should be valid. From the above considerations I conclude, that the explanatory power of the theory developed by Schneider (2005) is in fact higher than might be anticipated initially. We were not able to predict anything that was not already known from observations, but we can explain the observations (qualitatively) from external parameters: rotation rate, planetary radius, pole to equator temperature contrast, and a (not very well constrained) relation between the density scale height and optical thickness. 6 Comment on Diffusive Closure Most of the results rely on the assumption that eddy mixing occurs in a way that materially conserved quantities are mixed downgradient. This is usually true for passive tracers but is not necessarily true for active tracers like PV or potential temperature/entropy. However it turns out that under a wide range of external forcings the assumption holds. Fig. 5 shows the mixing efficiency (as measured by PV 6 We would have to modify our conclusions if, for instance, the albedo at the equator would be incredibly high, or some strong absorber were present in the polar tropopause region. 19

20 contour stretching) in an idealized GCM simulation: a well mixed layer near the surface exists, corresponding to the troposphere. To the extend that this well mixed region is also the region dominated by the isentropic overturning circulation, and as long as we consider only scales larger than typical eddy scales, the assumption appears to be justified. The existence of such a well mixed layer depends on surface drag and upward momentum fluxes (Greenslade and Haynes, 2008). Fig. 6 shows the zonal mean PV distribution in isentropic coordinates in the final state of a baroclinic life cycle integration (bottom). The initial tropopause height (indicative of planetary PV) and surface potential temperature are superimposed. It is evident that PV and surface potential temperature are well mixed and initial gradients are equilibrated by the baroclinic life cycle Comparison with Quasi Geostrophic Theory Now we can of course ask whether all this could have been derived from simpler models of already existing theories; the most well established competitor would be quasi geostrophic theory. Quasi geostrophic theory shares a number of assumptions with the isentropic theory: small Rossby number, stable stratification, and small aspect ratio (of vertical over horizontal scales); the latter is also implicit in the isentropic theory. Furthermore QG theory is formally analogous and the relations derived here have quasi geostrophic analogs. The major short coming of quasi geostrophic theory is its dependence on reference profiles of density and static stability (and the assumption that deviations are small). This also means the quasi geostrophic approximation assumes that the slope of isentropes with respect to pressure levels is small. Much of the theory developed here (and in the next section) hinges on the adjustment of static stability and the slope of isentropes, so that this is a major constraint. It is in a way obvious that a quasi geostrophic layer models are not able to predict 20

21 Figure 5: Atmospheric mixing patterns obtained from an idealized GCM integration (as measured by PV contour stretching). Mixing in the troposphere is homogeneous and generally high, while mixing in the stratosphere is lower and exhibits much more structure. (Haynes et al., 2001) 21

22 Figure 6: Zonal mean PV distribution in isentropic coordinates during the final state of a baroclinic life cycle experiment. The black line is the 2 PVU iso line, the white lines refer to zonal mean surface potential temperature and the red lines indicate the (WMO ) tropopause. Dashed lines refer to the initial state, solid lines to the baroclinically adjusted state; the initial tropopause location is indicative for the initial planetary vorticity distribution. Evidently the baroclinic wave equilibrated initial gradients and produces a well mixed troposphere and surface layer (if we ignore boundary artifacts), but there are also dynamical constraints on mixing, evident from the sharp drop in dynamical tropopause height (the 2 PVU iso line) at about 60 N. 22

23 fluxes associated with the intersection of isentropes with the ground: because perturbations to the reference profile have to remain small, the density can not vanish (and the reference density has a maximum near the surface). However continuous QG models pressure surfaces can intersect the ground and a surface potential temperature flux analogous the term in isentropic coordinates exists. Schneider (2005) further argues that continuous QG models typically do not support opposite signs of the PV gradient in different layers of the atmosphere. My interpretation of this is as follows: In the isentropic picture we were able to infer the sign of PV and surface potential temperature gradients from external parameters. This was possible because zonally averaged isentropic motion is essentially one dimensional and isentropic mass fluxes directly relate to entropy redistribution arguments. Because isentropes are only weakly sloped the contribution in the interior atmosphere would be limited to the β term and Ekman fluxes which we found to be fundamentally flawed. To the extend that quasi geostrophic dynamics is an approximation to isentropic dynamics, we can of course find an appropriate reference state and equivalent boundary conditions. 7 However here I have two objections: both PV gradient contributions (tropopause and surface) would be exported into the boundary conditions, in the form of potential temperature anomalies (as in the Eady model, Bretherton s Trick ). This would be necessary because the static stability and density have characteristic strong gradients at these boundaries. 8 First, both, the surface poten- 7 It may even turn out that the quasi geostrophic analogs of terms in the isentropic formulation will evaluate to similar values, even beyond their strict validity, however I would argue that this would be a consequence of their similarity to the more general theory; in the same way as we can derive Rossby waves from linear and nonlinear theory. 8 It would be possible to integrate these into the mean reference state, but the representation of advection would then be problematic because these quantities would vary strongly; also the inversion operator for PV would have to account for such gradients in the reference state, which would in a way change the theory and may be seen as unsatisfactory. 23

24 tial temperature flux as well as the surface layer PV flux are identical in QG theory, as Schneider (2005) already argued. Second, and more importantly, it would be difficult to justify a certain reference state and a set of boundary conditions without resorting to the isentropic picture. In quasi geostrophic theory we can of course assume a similar surface temperature gradient, but I am not aware of an argument that would justify a significant equatorward PV gradient in the near surface layer in quasi geostrophic theory. The only reasonable assumption would be a homogeneous poleward gradient due to the β effect, which is relatively small. Potential temperature anomalies at the upper boundary also correspond to PV anomalies ( Bretherton s Trick ), so one may argue that this produces the required gradients (as in the Eady model), and to the extend that quasi geostrophic theory is an approximation to the isentropic theory, it is of course correct, however, the lower boundary flux is then equivalent to the eddy flux of surface potential temperature, so that in any case one term is missing. Also I would argue that it is more difficult to justify these boundary conditions from first principles without resorting to empirical arguments or observations. My main point here is, that although formally analogous terms may exist in quasi geostrophic theory, we could not have inferred the qualitative features of entropy and mass transport in the atmosphere in the way we have done above (and as it was first done by Schneider, 2005). 3 The Thermal Stratification of the Troposphere and other Applications In this section I will summarize how the results from the previous section have been applied to infere some characteristics of the atmosphere that go somewhat beyond entropy 24

25 transport. Central will be the relationship between the tropopause height and the vertical extend of eddy mixing. 3.1 The Vertical Extend of Eddy Mixing Starting from the simplified form (12) of the isentropic mass flux balance, we can derive an expression for the vertical extend of eddy mixing by requiring that the overturning cell closes at some finite upper value of potential temperature. We will use the definition of isentropic density and integrate with respect to potential temperature in order to obtain an expression for the pressure level at which the isentropic mass flux closes, i.e. where the vertical integral of the isentropic mass flux vanishes. For this purpose we will again substitute diffusive closures for the eddy flux terms of surface potential temperature and PV: v sθ ss Ds y θs and v P D i y P. (18) Requiring that the mass flux closes at some isentrope denoted by θ max and neglecting the Ekman flux term, we write θmax θ min σv dθ θmax θ min D i y P P dθ + σ( θ s ) D s y θs 0 (19) Now we will replace the surface isentropic density by a static stability estimate σ( θ s ) ( σ( θ s )H( θ s θ s ) 2g p θ s) 1, and use the planetary vorticity approximation (13) to evaluate the PV gradient: θmax ( D i y σ σ β ) dθ D i f g θ min ( y p e θe β ) f [p e p s ] D s 2 g y θs p θ s. (20) Where the integral on the right hand side was evaluated, using σ = θ p/g, and the contribution of the pressure gradient at the surface y p θs was neglected, for it is generally 25

26 small; 9 the pressure gradient at the upper boundary is not necessarily small, however it turns out that it is at most 15% and usually below 5% of the contribution from the pressure difference (Schneider, 2006). We also assumed that the eddy diffusivity exhibits no essential vertical structure; further assuming D i = D s, we write p s p e f β y θs 2 p θ s (21) for the pressure difference between the surface and the upper boundary of the isentropic overturning cell. To the extend that the transport in mid latitudes is dominated by eddy fluxes (what we assumed), p e is the pressure level up to which eddies effect significant overturning and can possibly affect the thermal stratification. Schneider (2006) then goes on and defines the Bulk Stability v ; to an extend which will become clear in the next section it is a measure for the potential temperature difference between the tropopause and the surface. v = 2 p θ s ( p s p T P ) (22) p T P is the pressure at the tropopause. From this point he defines the Supercriticality S c in analogy to quasi geostrophic models of baroclinic instability as S c = p s p e p s p T P f β y θs v (23) and argues that the Supercriticality does not exceed unity S c 1 in any realistic atmosphere. There is a priori no strict argument to assume this but it appears to be supported by a large number of GCM experiments (see Fig. 8, right). The implication is that baroclinic eddies adjust potential temperature gradients and the tropopause pressure in such a way as to reduce the baroclinicity (i.e. the instability that causes them); heuristically this is of course reasonable. 9 This assumption should be generally valid in planetary atmospheres with a rigid lower boundary and surface friction (i.e. no slip condition). 26

27 Schneider (2006) further argues that this is also the reason why the inverse energy cascade does not go beyond the scale of the linearly most unstable mode (approximately the Rossby radius): because the Supercriticality (baroclinicity) does not significantly exceed unity, baroclinic eddies mostly remain in the weakly nonlinear regime. The inverse energy cascade is effected by nonlinear eddy eddy interaction, and is thus inhibited. 3.2 An Estimate for the Height of the Tropopause The Tropopause in a Nutshell 10 The tropopause is defined by the WMO as the height where the lapse rate falls below the threshold value of Γ T P < 2 K/km and the average over the next 2 km satisfies this criterion as well. This definition is purely empirical and gives virtually no hint at its physical significance. From a theoretical/dynamical point of view the tropopause is generally interpreted as the layer where dynamical adjustment of the lapse rate ceases, and radiative equilibrium dominates (cf. Held, 1982). Implicit in this view is a sense of statistical averaging, in particular, because the stratosphere is locally not in radiative equilibrium (only on average). In the light of the analysis presented above, it is straight forward, to associate dynamical adjustment with eddy fluxes in the isentropic overturning circulation, and define the height or pressure level at which the isentropic mass flux closes as the tropopause. However, this somewhat naive approach neglects the radiative contribution to the tropopause height. Fig. 7 illustrates the competing mechanism that determine the tropopause height. Because radiative equilibrium profiles in the lower troposphere are unstable with respect to convective (and baroclinic) overturning dynamical adjustment of the static stability occurs. This adjustment forces a certain lapse rate, which is not in radiative equilibrium anymore; heat is deposited in the troposphere (in accord with the cross isentropic 10 For a good review see Thuburn and Craig (2001), and references therein Schneider (2004) also discusses some aspect. 27

28 slope of mass flux streamlines in the poleward branch). The tropopause is then the point where the atmosphere attains a Temperature profile which is in radiative equilibrium. The latter is commonly referred to as radiative constraint and the former as dynamical constraint. A complication arises from the fact that heat flux convergence in the troposphere can destabilize the temperature profile further, which in turn affects the adjustment, so that radiative and dynamical adjustment have to be solved simultaneously. Two continuity conditions at the tropopause and two boundary conditions apply: one for the temperature and one for the radiative flux. Schneider (2004) proposes an alternate definition of the tropopause based on the isentropic mass flux: he defines the tropopause to be the isentropic level, at which 90% of the isentropic mass flux closes. Note however that this definition can only be evaluated in a meaningful way in a statistically averaged sense. A Dynamical Constraint To the extend that the static stability is adjusted by baroclinic eddies, a dynamical constraint for the tropopause height is given by the upper bound of the isentropic mass flux (21) p T P = p e. Using the definition of the bulk stability (22) we can write v = f β θ y s. (24) In order to estimate the height of the tropopause Schneider (2006) uses hydrostasy to approximate p θ s H z θ s / p s (with the scale height H = RT s /g) and writes the bulk stability as v 2 H z θ s p s p T P p s (25) Further assuming the static stability in the troposphere does not vary significantly with 28

29 Figure 7: A schematic illustrating the mechanisms that determine the tropopause height: solar insolation is received primarily at the surface and radiated to space at the top of the atmosphere; eddies transport heat toward the poles, and deposit heat in the troposphere in mid latitudes. The tropopause is thought to be at the height where radiative heat transfer begins to dominate over eddy fluxes. 29

30 height, we can write z θ s = ( θt P θ s ) /HT P, and the bulk stability takes the form v 2 H H T P p s p T P p s ( θt P θ s ). (26) In the earth atmosphere both fraction on the right hand side are somewhat smaller than unity, so that the factor in front of the potential temperature difference approximately evaluates to unity. It follows that for earth like atmospheres (i.e. where the tropopause height is somewhat higher than the scale height) the tropopause potential temperature can be estimate to θ T P θ s = f β θ y s. (27) If the scaled surface potential temperature gradient can be approximated by the equator to pole temperature contrast, we can further conclude, that an isentrope that is near the surface at the equator, will reach the tropopause near the poles. Fig. 8 (left) shows surface to tropopause potential temperature differences obtained from the scaled surface potential temperature gradient (27) plotted against values obtained from the tropopause definition of Schneider (2004) based on the isentropic mass flux streamfunction. The tropopause obtained using the definition of Schneider (2004) is indicated in Fig. 3 (right). The idealized GCM used to produce Figs. 3 & 8 is essentially the same. 4 Summary and Conclusion In this essay I have reviewed an attempt at describing the general circulation in terms of isentropic mass fluxes. Major theoretical progress was made by the description of isentropes intersecting the surface (due to Schneider, 2005). The theory falls short of giving a closed description of the global circulation. However we were able to gain theoretical insight into the mechanisms of mass and entropy fluxes in 30

31 Figure 8: Tropopause heights and tropospheric static stabilities obtained from multiple series of idealized GCM experiments. Left: the surface to tropopause potential temperature difference as a function of the scaled surface potential temperature gradient for different external parameter values. All values are close to the prediction (dotted line); the diamond represents the observed value for the earth. Right: the bulk stability vs. the scaled surface potential temperature gradient for different rotation rates and equator to pole temperature differences. The dashed line indicates a Supercriticality S = 1. The small crosses correspond to radiative convective equilibrium calculations (i.e. no dynamics), and are not actually observed. (Schneider, 2004, 2006) 31

32 different layers of the atmosphere. The major contributions are due to poleward eddy flux of PV in the interior atmosphere, and equatorward eddy flux of PV and surface potential temperature in the surface layer. The theory developed by Schneider (2005) permits an estimate for the height up to which the atmosphere is significantly affected by eddy fluxes, and Schneider (2004) uses this to derive a dynamical contraint for the tropopause height. The inferences were possible with only a small number of assumptions and are thus valid for a wide range of planetary atmospheres: stable stratification, fast rotation, and a rigid lower boundary with friction (i.e. no slip condition). This mostly excludes gas giants and slowly rotating planets. The highest uncertainty lies with the radiative forcing, which strongly depends on atmospheric composition and cloud layers. It would be an interesting exercise to work out equivalent inferences for forcing conditions radically different from those discussed by Schneider (2005) and observed on earth. As I argued (expanding on the arguments provided by Schneider, 2005) the theoretical progress would not have been possible without the framework of isentropic mass flux. Not only because of the limitation of quasi geostrophy to small perturbations about a reference state (for density and static stability), but also, and most importantly, because the formulation and theoretical justification of the assumption which eventually led to useful insights (PV gradient and mass flux attribution) would have been very difficult in any other framework. The reason for this is that the atmosphere is essentially driven by the redistribution of entropy and the zonal mean motion on isentropes is quasi one dimensional. The equations derived by Schneider (2005) have analog forms in quasi geostrophic theory, and it is possible that an evaluation of the corresponding QG terms in observational data will give similar results, even beyond their strict validity. However I would argue that this would be a consequence of their similarity to the more general isentropic formulation. Schneider (2004) also proposed a tropopause definition based on isentropic mass flux balance. Personally I hold the opinion that this is the physically most reasonable defini- 32

33 tion proposed to date. It is directly based on physical insight into the process that define the troposphere (and hence the tropopause). It is universally applicable, in any planetary atmosphere, and does not depend on specifics of parameters or boundary conditions (atmospheric composition or solar heating). I would argue that the latter is a consequence of the former. 33

34 References Greenslade, M. D., and P. H. Haynes (2008), Vertical Transition in Transport and Mixing in Baroclinic Flows, J. Atmos. Sci., 65, Haynes, P., J. Scinocca, and M. Greenslade (2001), Formation and maintainance of the extratropical tropopause by baroclinic eddies, Geophys. Res. Lett., 28 (22), Held, I. M. (1982), On the Height and the Static Stability of the Troposphere, J. Atmos. Sci., 39, Held, I. M., and V. D. Larichev (1996), A Scaling Theory for Horizontally Homogeneous, Baroclinically Unstable Flow on a Beta Plane, J. Atmos. Sci., 53, Held, I. M., and T. Schneider (1999), The Surface Branch of the Zonally Averaged Mass Transport Circulation in the Troposphere, J. Atmos. Sci., 56, Kallenberg, P., P. Berrisford, B. Hoskins, A. Simmons, S. Uppala, S. Lamy-Thepaut, and R. Hine (2005), ERA 40 Atlas, ECMWF. Pauluis, O., A. Czaja, and R. Korty (2008), The Global Atmospheric Circulation on Moist Isentropes, Science, 321, Schneider, T. (2004), The Tropopause and the Thermal Stratification in the Extratropics of a Dry Atmophere, J. Atmos. Sci., 61, Schneider, T. (2005), Zonal Momentum Balance, Potential Vorticity Dynamics, and Mass Fluxes on Near Surface Isentropes, J. Atmos. Sci., 62, Schneider, T. (2006), Self Organization of Atmospheric Macroturbulence into Critical States of Weak Nonlinear Eddy Eddy Interaction, J. Atmos. Sci., 63,

35 Sun, D.-Z., and R. S. Lindzen (1994), A PV View of the Zonal Mean Distribution of Temperature and Wind in the Extratropical Troposphere, J. Atmos. Sci., 51 (5), Thuburn, J., and G. C. Craig (2001), Stratospheric Influence on Tropopause Height: Radiative Constraint, J. Atmos. Sci., 57,