PHY2504 Course Project: Zonal Momentum Balance, Entropy ...

PHY2504 Course Project:

Zonal Momentum Balance, Entropy

Transport, and the Tropopause

Andre R. Erler

April 21th, 2009

1

Andre R. Erler PHY2504 April 23, 2009

Contents

1 Introduction 3

2 Zonal Momentum Balance in Isentropic Mass Flux 4

2.1 Zonally Averaged Momentum Balance and Boundary Conditions . . . . . . 7

2.2 Eddy Fluxes and the Mean Isentropic Circulation . . . . . . . . . . . . . . 12

2.2.1 Verification of the Predictions . . . . . . . . . . . . . . . . . . . . . 15

2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.1 Comparison with Quasi–Geostrophic Theory . . . . . . . . . . . . . 20

3 The Thermal Stratification of the Troposphere and other Applications 24

3.1 The Vertical Extend of Eddy–Mixing . . . . . . . . . . . . . . . . . . . . . 25

3.2 An Estimate for the Height of the Tropopause . . . . . . . . . . . . . . . . 27

4 Summary and Conclusion 30

2


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

1This 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


coordinates (top); there are three major overturning cells: the Hadley cell, which is domi-

nated 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 corre-

sponding 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 isen-

tropes 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


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


Figure 2: Top panel: the annually averaged mass flux stream–function in isentropic coor-

dinates. 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 circula-

tion branch occurs on isentropes which frequently intersect the surface. Bottom panel:

significant cross–isentropic flow occurs on near surface isentropes, while diabatic heat-

ing/cooling in the interior atmosphere is small. (Kallenberg et al., 2005)

6


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 = −1g∂θp, which is essentially the

mass (∼ g∆p) between isentropic layers. Isentropic mass conservation in flux form then

takes the form

∂tσ + ∂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(σu2) + ∂y(σuv) + ∂θ(σuQ) = −σ∂xM + 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 + cpT assuming the role of

7


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 θs 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 < θs) 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 − θs) , ∀ θ . (3)

Equations (1) and (2) can now be combined; using definition (3) and with ζθ = ∂yu− ∂xv

denoting the relative vorticity perpendicular to isentropic surfaces, we arrive at

[∂tu− (f + ζθ) v]σH(θ − θs) = [−∂xB −Q∂θu+ F x]σH(θ − θs) . (4)

B = ((v2 + u2)/2 + gz + cpT ) is the Bernoulli function, which essentially represents the

total energy of the fluid at a given location (in stationary flow B is constant along stream-

lines).

Note that both sides of the equality (4) are multiplied by the term σH(θ − θs), so that

σ cancels in the interior atmosphere (θi > θs ∀ t), where isentropes do not intersect the

ground (and σ > 0 ∀ θi).

∂tu− (f + ζθ) v = −∂xB −Q∂θu+ F x ∀ θ > θs (5)

8


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

x1φ(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 ∂xφ(x) = [φ(x)]x2

x1= 0, so that in the interior atmosphere

zonal and temporal averaging reduces equation (5) to

vP∗

+ JQ∗

+ JF∗

= 0 . (6)

P = (f + ζθ) /σ is the potential vorticity on isentropes, and JQ = Q∂θu/σ and JF = F x/σ

are the diabatic terms. Note that P , JQ, and JF 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 = ps , ∀θ < θs, is set to the instantaneous surface pressure

• Velocity (u, v) = 0 , ∀ θ < θs vanishes, in accord with the no–slip boundary condition

• Relative vorticity ζθ = 0 , ∀ θ < θs also vanishes (Convention I in Schneider , 2005)2

2An alternative convention also discussed by Schneider (2005) sets the absolute vorticity σP = f+ζθ = 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 ζθ = −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 < θs.

9


• The diabatic terms are also set to zero: JQ ⇒ JQH(θ − θs) and JF ⇒ JFH(θ − θs)

Note that the mean of the step–function can assume a non–integer value 0 ≤ H(θ − θs) ≤ 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 [u2 + v2]x2

x1= 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:

∂xBH(θ − θs) =1

∆x

∫ x2

x1

B(x)H (θ − θs(x)) dx

=1

∆x[BH(θ − θs)]x2

x1− 1

∆x

∫ x2

x1

B∂xH(θ − θs) dx

= Mδ(θ − θs)∂xθs . (7)

The step–function H(θ − θs) vanishes at the boundary (x1, x2), so only the integral term

survives (which is equivalent to a zonal average). With ∂xH(θ − θs(x)) = δ(θ − θs)∂xθs,

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


interior as well as the surface layer now reads

σvP∗

+ σJQ∗

+ σJF∗

= Mδ(θ − θs)∂xθs . (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 ′

∗+ σJQ

∗+ σJF

∗)

+1

P∗ Mδ(θ − θs)∂xθs ; (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 (θs)

∗)−1

and introduce

the balanced surface eddy velocity v′s = f−1∂xM′s:

3∫ θint

θmin

M∂xθsδ(θ − θs)P∗ dθ ≈ −1

P (θs)∗

∫ θint

θmin

∂xM θsδ(θ − θs) dθ ≈ −f

P (θs)∗ v′sθ

′s

s. (10)

The lower boundary of integration θmin is the lowest potential temperature that occurs

at the surface and the upper boundary θint 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

θmin

σv∗ dθ ≈ −∫ θint

θmin

σv′P ′∗

+ σJQ∗

+ σJF∗

P∗ dθ − f

P (θs)∗ v′sθ

′s

s. (11)

3Schneider (2005) gives a slightly different definition: v′s = f−1∂x(M ′s − cpθ′s), presumably to draw an

analogy to quasi–geostrophic theory. The modification is of no consequence since the term ∂xθ′s drops

out after averaging: θ′s∂xθ′s = ∂x(θ′sθ′s) = 0.

11


Or, after neglecting diabatic heating JQ∗

and using the planetary vorticity approximation

P∗ ≈ f/σ (valid at small Rossby number), a simpler form can be written as∫ θint

θmin


θmin

σv′P ′∗

+ σJF∗

P∗ dθ − σ(θs)v′sθ

′s

s. (12)

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


number (fast rotation), and stable stratification with respect to adiabatic vertical over-

turning.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

∂yP∗

= ∂y

(f

σ

)=β

σ− f ∂yσ

σ2. (14)

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

θmin


θmin

σv′P ′∗

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

θmin


θmin

σ∂yP∗

P∗ dθ ≈ −

∫ θint

θmin

Di

(∂yσ − σ

β

f

)dθ , (16)

where Di 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

4In convectively dominated atmospheres (e.g. those of gas giants) isentropic coordinates loose their ap-

plicability.

13


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 ∂yθs is obviously directed

equatorward, and this holds for most planetary atmospheres.

With the PV–gradient (14), we can write∫ θint

θmin


θmin

(Di ∂yσ −Di σ

β

f+σ2

fJF∗)dθ + σ(θs)Ds∂yθs , (17)

where Di and Ds 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(θs)σ(θs),

where Π(θs) 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

5Infact both estimates, the PV–gradient and the gradient of θs, depend on the distribution Π(θs) of

14


2.2.1 Verification of the Predictions

Schneider (2005) conducted highly idealized GCM experiments and computed the isen-

tropic mass flux, corresponding to the theory developed above.

The GCM Schneider (2005) used consisted only of a dynamical core, without any so-

phisticated 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 tem-

perature 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


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


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 10th and 90th percentile

(dotted lines). (Pauluis et al., 2008, NCEP–NCAR Reanalysis data)

17


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 la-

tent 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 gradi-

ent 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 ter-

restrialplanets 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


towards the pole. This may appear arbitrary but is in fact also very generic for terrestrial-

type 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 coordi-

nates 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 signifi-

cantly 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–

6We 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


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.

2.3.1 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 hori-

zontal 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 pro-

files 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


Figure 5: Atmospheric mixing patterns obtained from an idealized GCM integration (as

measured by PV–contour stretching). Mixing in the troposphere is homogeneous and gen-

erally high, while mixing in the stratosphere is lower and exhibits much more structure.

(Haynes et al., 2001)

21


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 distri-

bution. 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


fluxes associated with the intersection of isentropes with the ground: because perturba-

tions 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 pres-

sure 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 tem-

perature 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 contri-

bution 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 refer-

ence state and equivalent boundary conditions.7 However here I have two objections: both

PV–gradient contributions (tropopause and surface) would be exported into the bound-

ary 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-

7It 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.8It 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


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 gra-

dient, 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 rel-

atively 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 ap-

plied to infere some characteristics of the atmosphere that go somewhat beyond entropy

24


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′s

s ≈ −Ds ∂yθs and v′P ′∗ ≈ −Di ∂yP

∗. (18)

Requiring that the mass flux closes at some isentrope denoted by θmax and neglecting the

Ekman flux term, we write∫ θmax

θmin

σv∗ dθ ≈∫ θmax

θmin

Di ∂yP∗

P∗ dθ + σ(θs)Ds∂yθs ≈ 0 (19)

Now we will replace the surface isentropic density by a static stability estimate σ(θs) ≈

σ(θs)H(θs − θs) ≈ −(

2g∂pθs)−1

, and use the planetary vorticity approximation (13) to

evaluate the PV–gradient:∫ θmax

θmin

Di

(∂yσ − σ

β

f

)dθ ≈ Di

g

(∂ype|θe −

β

f[pe − ps]

)≈ Ds

2 g

∂yθs

∂pθs . (20)

Where the integral on the right–hand side was evaluated, using σ = −∂θp/g, and the

contribution of the pressure gradient at the surface ∂yp|θs was neglected, for it is generally

25


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 Di = Ds, we write

ps − pe ≈f

β

∂yθs

2 ∂pθs (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), pe 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 ∂pθs

(ps − pTP ) (22)

pTP is the pressure at the tropopause. From this point he defines the Supercriticality Sc in

analogy to quasi–geostrophic models of baroclinic instability as

Sc =ps − peps − pTP

≈ −fβ

∂yθs∆v

(23)

and argues that the Supercriticality does not exceed unity Sc ≈ 1 in any realistic atmo-

sphere. 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 baro-

clinic 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.9This assumption should be generally valid in planetary atmospheres with a rigid lower boundary and

surface friction (i.e. no–slip condition).

26


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 ΓTP < 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 tro-

popause 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 sta-

bility occurs. This adjustment forces a certain lapse–rate, which is not in radiative equi-

librium anymore; heat is deposited in the troposphere (in accord with the cross–isentropic

10For a good review see Thuburn and Craig (2001), and references therein Schneider (2004) also discusses

some aspect.

27


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 con-

straint. 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 isen-

tropic 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) pTP = pe. Using the definition of the bulk stability (22) we

can write

∆v =f

β∂yθs . (24)

In order to estimate the height of the tropopause Schneider (2006) uses hydrostasy to

approximate −∂pθs ≈ H∂zθ

s/ps (with the scale height H = RT

s/g) and writes the bulk

stability as

∆v ≈ 2H ∂zθs ps − pTP

ps(25)

Further assuming the static stability in the troposphere does not vary significantly with

28


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


height, we can write ∂zθs

=(θTP − θs

)/HTP , and the bulk stability takes the form

∆v ≈ 2H

HTP

ps − pTPps

×(θTP − θs

). (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

θTP − θs =f

β∂yθs . (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 stream-

function. 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


Figure 8: Tropopause heights and tropospheric static stabilities obtained from multiple se-

ries 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 dia-

mond 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


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 refer-

ence 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 re-

distribution 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


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 (at-

mospheric composition or solar heating). I would argue that the latter is a consequence of

the former.

33


References

Greenslade, M. D., and P. H. Haynes (2008), Vertical Transition in Transport and Mixing

in Baroclinic Flows, J. Atmos. Sci., 65, 1137–1157.

Haynes, P., J. Scinocca, and M. Greenslade (2001), Formation and maintainance of the

extratropical tropopause by baroclinic eddies, Geophys. Res. Lett., 28 (22), 4179–4182.

Held, I. M. (1982), On the Height and the Static Stability of the Troposphere, J. Atmos.

Sci., 39, 412–417.

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, 946–952.

Held, I. M., and T. Schneider (1999), The Surface Branch of the Zonally Averaged Mass

Transport Circulation in the Troposphere, J. Atmos. Sci., 56, 1688–1697.

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, 1075–1078.

Schneider, T. (2004), The Tropopause and the Thermal Stratification in the Extratropics

of a Dry Atmophere, J. Atmos. Sci., 61, 1319–1339.

Schneider, T. (2005), Zonal Momentum Balance, Potential Vorticity Dynamics, and Mass

Fluxes on Near–Surface Isentropes, J. Atmos. Sci., 62, 1884–1900.

Schneider, T. (2006), Self–Organization of Atmospheric Macroturbulence into Critical

States of Weak Nonlinear Eddy–Eddy Interaction, J. Atmos. Sci., 63, 1569–1586.

34


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), 757–

772.

Thuburn, J., and G. C. Craig (2001), Stratospheric Influence on Tropopause Height: Ra-

diative Constraint, J. Atmos. Sci., 57, 17–28.

35

PHY2504 Course Project: Zonal Momentum Balance, Entropy ...

Documents

Transcript of PHY2504 Course Project: Zonal Momentum Balance, Entropy ...