The General Circulation of the Atmosphere

3/4/19

Summary from last lecture

• Derived angular-momentum budget for the atmosphere The general circulation • Above the boundary layer, and under QG scaling: � �� = �� of the atmosphere �� Section II: The angular-momentum budget • Eddy-flux divergence determines mean meridional circulation

Maintenance of a barotropic jet • In column integral, momentum convergence balanced by frictional (and form drag) torques

• Implies westerlies in region of momentum convergence

• Observed distribution of eddy fluxes has strong convergence in midlatitude upper-troposphere

Northward eddy-momentum fluxes transient eddy flux of momentum

Peixoto & Oort Peixoto & Oort

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transient eddy transient eddy momentum flux momentum flux divergence/convergence divergence/convergence streamfunction

Peixoto & Oort Peixoto & Oort

transient eddy transient eddy momentum flux momentum flux divergence/convergence divergence/convergence vertical surface vertical momentum winds momentum flux flux

Peixoto & Oort Peixoto & Oort

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Angular-momentum cycle

Peixoto & Oort Peixoto & Oort

Atmospheric angular momentum Schematic of momentum fluxes cycle • The preceding analysis paints a picture in which angular momentum is converged into the upper troposphere in midlatitudes, where it is transported downwards and removed at the

height surface by friction/mountain torques (westerly winds) • This angular momentum is primarily drawn from subtropical latitudes, where it is provided by friction with the Earth’s surface (easterly winds) EQUATOR 30S 60S POLE • Note that this angular-momentum transport is up North gradient!

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Total angular momentum of the atmosphere

�� + � ⋅ (�� ) = − + �� cos �(�� + � ). ��

Integrate over whole atmosphere,

� �� = − � �� + � ��. ��

change in angular form drag friction momentum

Atmosphere exchanges angular momentum with the Earth through form drag and friction. Reflected in changes to the length of day.

Summary Barotropic vs Baroclinic

• The mean meridional circulation in the extratropics is determined by the These pieces of useful jargon come from the vorticity equation: distribution of eddy momentum fluxes in the upper troposphere �� 1 1 = � ⋅ �� − � � ⋅ � + ��×�� + �×� �� • The surface winds are determined by the vertically integrated eddy flux convergence or divergence into a given latitude band tilting stretching baroclinic friction

• Thus in a quasi-geostrophic atmosphere, barotropic jets are can only be A barotropic fluid has no baroclinic production of vorticity. maintained in the presence of an anisotropic distribution of eddies This requires that isobars and isopycnals are parallel • What determines the distribution of eddy-momentum fluxes in Earth’s atmosphere? One way of achieving this is if pressure is a function of density only

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Barotropic vs Baroclinic Barotropic vs Baroclinic

Now, for a fluid that obeys quasi-geostrophic scaling, the flow For an ideal gas, will approximately satisfy the thermal wind relation,

� � �� ×�� = ��×�� − ��×�� = �×��

where � is the horizontal wind. Thus, an ideal gas is barotropic if isotherms are parallel to isobars. So horizontal temperature gradients are associated with vertical gradients in the wind.

That is, if there are no horizontal temperature A barotropic atmosphere therefore has depth-independent gradients (when working in pressure coordinates) flow

Barotropic vs Baroclinic Barotropic vs Baroclinic

More generally, we think of Sometimes you will hear the term “equivalent the depth-independent 2nd baroclinic barotropic component of the flow as barotropic” the “barotropic component”.

1st This originates in analytic 3rd baroclinic baroclinic This means that the shear is always in the same models of the atmosphere direction as the wind (The wind does not turn with based on a finite number of modes. The gravest mode is height) the barotropic mode, the next mode is the “1st baroclinic mode” etc.

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The vorticity equation The vorticity equation

We begin by considering the equation for vorticity, a quantity related to angular To begin with, we restrict ourselves to a single layer fluid with constant density so momentum. that the velocity does not depend on height, and there is no vertical velocity.

In such a fluid, the density is constant � so that the continuity equation, In general, the vorticity is defined by the curl of the velocity, � = �×� �� + � ⋅ �� = 0 We are interested in the radial component of the vorticity, using the standard �� formula for curl in spherical coordinates, this may be written, reduces to horizontal non-divergence of the flow: 1 �� cos � � = − � cos � �� ⋅ � = 0 In the thin shell approximation, we neglect variation in � in the above equation so Or in expanded form: that,

1 �� cos � 1 �� cos � + = 0 � = − � cos � �� cos � ��

The vorticity equation The vorticity equation We now cross differentiate the momentum equations, that is, we apply the following transformation: Consider our usual primitive equation set:

1 � �� 1 �� 1 � � + � cos � − � � + � = − − + � �� 1 �� cos � �� cos � �� cos � �� 2 = 2Ω sin � � + tan � − + � �� cos � �� 1 � �� 1 �� 1 � � + � + � � + � = − − + � � cos � �� 2 �� 1 �� = −2Ω sin � � − tan � − + � �� After some algebra, this gives us an equation for vorticity, which may be written, With a bit of rearrangement, we can express these as,

� � + � = − � + � � ⋅ � + �×� �� 1 �� 1 � � + � �� − � � + � = − − + � �� cos � �� cos � �� 2

where the subscript ℎ refers to vectors with no vertical component. By the continuity equation for a single layer fluid, the divergence term on the right-hand side vanishes. Furthermore, if we restrict ourselves to inviscid flow, �� 1 �� 1 � � + � we have simple conservation of absolute vorticity: + � � + � = − − + � �� 2 � � + � = 0 �� where we have omitted terms depending on � due to our single layer assumption.

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Absolute vorticity Kelvin’s circulation theorem

Absolute vorticity is the sum of the planetary and For this homogenous fluid, we also have that the relative vorticity circulation around any material contour is conserved:

12.1 Maintenance of a Barotropic Jet 511 DΓ For a fluid at rest, the absolute vorticity increases = 0 monotonically with latitude.0 ��

More generally, the absolute vorticity is dominated 500 Γ = � ⋅ �� = � + � �� by the planetary component (outside the deep Height (mb) tropics), and so � + � increases with latitude for 1000 realistic flows-80 -40 0 40 80 -80 -40 0 40 80 Latitude Latitude

Fig. 12.1 The time-averaged zonal wind at 150° W (in the mid Pacific) in December-January February (DFJ, left), March-April-May (MAM, right). The 1 contour interval is 5 m s . There is a double jet in each hemisphere, one in the subtropics and one in midlatitudes, especially apparent in the right panel. The subtropical jets are in thermal wind balance with a strong meridional temperature gradient at the subtropical edge of the Hadley Cell, whereas the midlatitude jets have a stronger barotropic component, and are associated with eddy momentum flux convergence and westerly winds at the surface. Apply Kelvin’s circulation theorem where !ia and uia are the initial absolute vorticity and velocity, ui is the initial zonal Westerlies and circulation velocity in the earth’to a polar “cap”s frame of reference, and the line integrals are around the line of latitude. For simplicity let us take ui 0 and suppose there is a disturbance equator- D wards of the polar cap, and that this results in a distortion of the material line around the latitude circle C (Fig. 12.2). Since we are supposing the source of the disturbance • Assume atmosphere initially stationary, absolute is distant from the latitude of interest, then if we neglect viscosity the circulation along vorticity increases northwards • Deformation of the contour causes flux of vorticity out of the polar cap • By Kelvin’s circulation theorem, this implies decrease of zonal wind speed poleward of stirring • Apply same logic to South polar cap: easterlies form everywhere except the sirred region • By conservation of angular momentum; westerlies must appear under the stirred region! Fig. 12.2 The eects of midlatitude disturbance. If initially the absolute vorticity increases monotonically polewards, then the disturbance will bring • Requires irreversible mixing away from the region of fluid with lower absolute vorticity into the cap region. Then, using Stokes stirring theorem, the velocity around the latitude line C will become more westward.

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Rossby waves and momentum Define a streamfunction fluxes

Continue with the non-divergent homogenous fluid We then have, Consider vorticity equation under tangent plane approximation: �′ = � Ψ

�� Searching for exponential solutions: + � + � + � = 0 �� Ψ = �� = �� We can derive the Rossby wave dispersion relation: �� = �� − Linearise about basic state � = (� � , 0): � + � ��′ �� + �(�) + �′ � − = 0 Where we have assumed the mean zonal velocity is constant so that �� = 0.

Group velocity gives the direction momentum is converged toward of energy propagation the wave source �� 2� � = = �� k + � angular momentum flux

Also, note that the momentum flux wave wave activity flux source � �� ′ = − 2

The group velocity is in the opposite direction to the momentum flux!

8 12.1 Maintenance of a Barotropic Jet 515

Figure 12.3 Pseudomomentum stirring, which in reality occurs via baroclinic instability, is con- ﬁned to midlatitudes. Because of Rossyby wave propagation away from the source region, the distribution of pseudomomentum dissipation is broader, and the sum of the two leads to the zonal wind distribution shown, with positive (eastward) values in the region of the stirring. See also Fig. 12.8. will not locally balance in the region of the forcing, producing no net winds. That can only occur if the dissipation is conﬁned to the region of the forcing, but this is highly unlikely because Rossby waves are generated in the forcing region, and these propagate 3/4/19 meridionally before dissipating, as we now discuss.

12.1.4 III. Rossby waves and momentum flux We have seen that the presence of a mean gradient of vorticity is an essential ingredient in the mechanism whereby a mean flow is generated by stirring. Given such, we expect Rossby waves to be excited, and we now show how Rossby waves are intimately related to the momentum flux maintaining the mean flow. Kinematics of momentum If a stirring is present in midlatitudes then we expect that Rossby waves will be generated there. To the extent that the waves are quasi-linear and do not interact then transport

12.1 Maintenance of a Barotropic Jet 517

Figure 12.5 The momentum transport in physical space, caused by the propagation of Rossby waves away from a source in midlatitudes. The ensuing bow- shaped eddies are responsible for a convergence of momentum, as indicated in the idealization pic- tured. y↑ →x

* The radiation condition and Rayleigh friction Fig. 12.4 Generation of zonal flow on a ˇ-plane or on a rotating sphere. A common trick in fluid dynamics, especially in problems of wave propagation, is to add a small amount of friction to the inviscid problem.3 The solution of the ensuing Stirring in midlatitudes (by baroclinic eddies) generates Rossby waves that problem in the limit of small friction will often make clear which solution is physi- propagate away from the disturbance. Momentum converges in the region cally meaningful in the inviscid problem, and therefore which solution nature chooses. Consider the linear barotropic vorticity equation with linear friction, of stirring, producing eastward flow there and weaker westward flow on its @✏ @ ˇ r✏ (12.26) flanks. @t C @x D where r is a small friction coefficient. The dispersion relation is ˇk ! i r !R.k; l/ i r; (12.27) D K2 D

where !R is deﬁned by (12.21), and the wave decays with time. Now suppose a wave is generated in some region, and that it propagates meridionally away, decaying as moves away. Then, instead of an imaginary frequency, we may suppose that the frequency is real and the y-wavenumber is imaginary. Speciﬁcally, we take l l0 l0 where l0 2 1=2 D C D Œˇ=.u c/ k . for some zonal wavenumber k, as in (12.23), and ! !R.k; l0/. ˙ D For small friction, we obtain l0 by Taylor-expanding the dispersion relation around its inviscid value, !R.k; l0/, giving

@!R.k; l0/ ! i r !R.k; l/ !R.k; l0/ l0; (12.28) C D ⌦ C @l and therefore i r l0 y (12.29) D cg y where cg @ !R.k; l0/ is the y-component of the group velocity. The wavenumber is D l imaginary, so that the wave either grows or decays in the y-direction. The wave solution

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Rossby wave propagation Randel & Held (1990)

Waves can only propogate if � is real

�� = �� − � + �

This requires that

� > � � − �

In particular, the phase velocity must be smaller than the mean flow

� − � > 0

• This simple barotropic model captures the essense of the wave-mean flow interaction in the upper troposphere at midlatitudes • In reality, the eddies are produced by baroclinic instability, and a model with at least two layers is required • Ch 12 of Vallis (2006) and the Held lecture notes provide a detailed discussion of how to extend this model to multiple layers, and continuous stratification • Since the eddy propagation depends on the mean flow, the potential for feedbacks exist • Does such a feedback lead to the sharpening of the jet? • How do these feedbacks change the response of the jet to perturbations (e.g., annular modes)? • How do these feedbacks work locally (blocking?)