Equilibration in an Eddy Resolving Model with Simplified ... · Equilibration in an Eddy Resolving Model with Simplified Physics* AMY SOLOMON International Pacific Research Center,

15 MARCH 2001 561S O L O M O N A N D S T O N E

q 2001 American Meteorological Society

Equilibration in an Eddy Resolving Model with Simplified Physics*

AMY SOLOMON

International Pacific Research Center, SOEST, University of Hawaii at Manoa, Honolulu, Hawaii

PETER H. STONE

Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology,Cambridge, Massachusetts

(Manuscript received 6 July 1999, in final form 28 June 2000)

ABSTRACT

The role of waves in maintaining the midlatitude tropospheric climate is investigated in a dry high-resolutionquasigeostrophic b-plane channel model coupled to both a simplified model of the atmospheric boundary layerand an interactive static stability.

The climate of the model’s equilibrated state is found to be separated into two dynamical regimes, one withinthe boundary layer and the other within the free troposphere. Thermal diffusion in the atmospheric boundarylayer prevents the eddies from modifying the mean temperature structure there by damping temperature fluc-tuations. The potential vorticity gradients are essentially eliminated in the lower troposphere above the boundarylayer, in agreement with observations. The homogenization of potential vorticity occurs in the region where thebaroclinic waves have a critical layer, and is accomplished mainly by an increase in the static stability in thelower troposphere due to the vertical eddy heat fluxes.

Even though the model has kinetic energy and enstrophy spectra characteristic of a fully turbulent flow, theequilibrated state of the model is essentially maintained by wave–mean flow interaction, primarily by theinteraction between wave 5 and the zonal mean state. The zonal mean of the equilibrated state is found to belinearly stable to all waves. The largest-scale wave in the fully nonlinear state, wave 4, is found to be maintainedby an energy cascade from the higher wavenumbers. However when wave 4 is large, stability analysis indicatesthat it is unstable, with the growing mode dominated by wave 6. This instability appears to saturate quickly andhand its energy over to wave 5. The result is that the amplitude of waves 4 and 5 in the equilibrated state arestrongly anticorrelated, but the fluctuations in total eddy kinetic energy are strongly correlated with the fluctuationsin the sum of the energy in waves 4 and 5.

1. Introduction

The Charney–Stern criterion for instability is a nec-essary condition for instability but it is not sufficient.A basic state that has a change of sign in the meridionalgradient of potential vorticity may be stable. A linearstability analysis of the equilibrated state proposed byLindzen (1993) shows that the baroclinic normal modeshave been stabilized even though this state still satisfiesthe Charney–Stern necessary condition for instability.

The equilibrated state proposed by Lindzen (1993) is

* School of Ocean and Earth Science and Technology PublicationNumber 5308 and International Pacific Research Center PublicationNumber 56.

Corresponding author address: Dr. Amy Solomon, InternationalPacific Research Center, SOEST, University of Hawaii at Manoa,1000 Pope Road, Honolulu, HI 96822.E-mail: [email protected]

consistent with the observational studies of Fullmer(1982), Sun and Lindzen (1994), and Morgan (1994),which have shown that the potential vorticity gradientsin the extratropics are significantly less than b, the me-ridional gradient of planetary vorticity, in the midtro-posphere but large in the planetary boundary layer anda maximum at the tropopause. Stone and Nemet (1996)have argued that this vertical structure is consistent witha layer of strong baroclinic adjustment in the lower tro-posphere, separating layers with relatively little baro-clinic adjustment in the boundary layer and in the uppertroposphere. In these studies it was found that the ob-served extratropical temperature structure is determined,in part, by the mixing of potential vorticity along is-entropic surfaces and that the impact of this mixingvaries considerably in the vertical.

In order to resolve realistic neutral states, then, itappears that it is necessary to allow the dynamical bal-ance at the surface, in the midtroposphere, and at thetropopause to vary. A two-level model couples the dy-namics in the lower troposphere with the upper tropo-

562 VOLUME 58J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

sphere. This causes the dynamics at the jet and the dy-namics in the boundary layer to interact in a very ar-tificial way, and the equilibrated state of the two-levelmodel may have very little to do with the observedclimate. If the dynamics in the observed climate areroughly approximated by the state hypothesized byLindzen (1993), a model with more than two levelswould be necessary to resolve it.

This process model study attempts to construct a mod-el adequate to resolve a neutral state that is relevant tothe atmosphere. This model is used to illustrate the im-portance of the coupling between the quasigeostrophicdynamics in the free troposphere and the dynamics inthe atmospheric boundary layer (ABL) in maintenanceof the equilibrated state. Since the interaction betweensurface heat fluxes, static stability, and vertical eddyheat fluxes makes an important contribution to the eddy–mean flow interaction (Stone 1972; Gutowski et al.1989), we include an interactive static stability in ourmodel. The extent to which potential vorticity is ho-mogenized is explored and discussed in light of currenttheories about the nonlinear equilibration of the ob-served midlatitude troposphere. This issue is motivatedby the study of Lindzen (1993), which hypothesized thata possible simplistic equilibrated state that describes theobserved midlatitude troposphere is a state where thepotential vorticity gradients in the midtroposphere havebeen homogenized such that waves propagating alongthe tropopause and the surface of the earth can no longerinteract.

Section 2 describes the model used in this study. Sec-tion 3 demonstrates the adequacy of the model’s reso-lution and domain size. Section 4 compares the equil-ibrated state of the model to observations. Section 5discusses the evolution of the model’s climate to theequilibrated state. Section 6 discusses the role of wave–wave and wave–mean flow interactions in the equili-brated state. Section 7 discusses the stability character-istics of the equilibrated state. Section 8 discusses theimplications of these results for wave–mean flow inter-action in equilibrated states similar to the atmosphere’s.

2. The model

The model used in this study integrates the quasi-geostrophic equations coupled to an equation for thehorizontally averaged potential temperature on a mid-latitude b plane. The model has variable vertical andhorizontal resolution. The zonal, meridional, and ver-tical coordinates are evaluated in gridpoint space. Gridpoints in the vertical are equally spaced as a functionof pressure. There is a rigid lid in pressure coordinatesat the top and bottom of the model. There is no flowthrough the channel walls, and periodic boundary con-ditions are applied to all fields in the zonal direction.

All runs are initially in radiative–convective equilib-rium (RCE). The model is run for 20 days without aperturbation to allow the dynamics in the planetary

boundary layer to approach an equilibrium consistentwith RCE. Then the model is perturbed with small-amplitude waves 1–9. The amplitude of the pressureperturbation at the surface is 10 Pa in each of the ninewaves.

The standard run (SR) has a channel width of 10 000km and a channel length of 21 040 km. The channelwidth is approximately 908 in latitude and was chosenin order to prevent the dynamics at the channel wallsfrom modifying the dynamics at the center of the chan-nel. The horizontal resolution is equal in both the zonaland meridional directions and is set equal to 300 km inthe standard run. The standard run has 17 equally spacedvertical levels.

a. Model equations

This model integrates the quasigeostrophic potentialvorticity equation in pressure coordinates:

˙]q ] QR5 2J(c, q) 2 f 1 k · = 3 F,0]t ]p spcp

where R is the ideal gas constant, cp is the specific heatat constant pressure, and s is the static stability param-eter, and

] 1 ]c2 2q 5 ¹ c 1 by 1 f ,0 ]p s ]p

where c is the geostrophic streamfunction and b is themeridional gradient of planetary vorticity evaluated at458. The Q represents diabatic heating. The F representsfrictional dissipation and has two contributions: a ¹6

diffusion to remove vorticity at the smallest scales, anda boundary layer contribution described in section 2f.

The quasigeostrophic equations are coupled to anequation for the static stability tendency. This equationis derived from the horizontally averaged thermody-namic equation. The vertical potential temperature flux-es by the zonal mean flow are assumed to be insignif-icant relative to the eddy fluxes. This assumption canbe justified by scaling analysis (Gutowski 1983), and isconsistent with two-level model results (Zhou and Stone1993).

The horizontally averaged potential temperature ten-dency at time step t is given by

R /cp˙] ] Q pt 0[u ] 5 2 [v9u9] 1 ,t t t 1 2[ ]]t ]p c pp

where p0 is the surface pressure and Qt is the diabaticheating at time step t. Primes are used to designate de-viations from the zonal mean and [ ] indicate a globalareal mean. The static stability parameter is calculatedfrom the updated mean potential temperature.

b. Surface air tendency equation

An equation for the surface air temperature tendencyis integrated assuming that the fluxes of heat out of the


sea are convergent into the air just at the surface. Thevertical wind is assumed to be equal to zero. The seatemperature is held fixed. The turbulent fluxes from thesurface air into the first model level are calculated usinga flux diffusion approximation. The equation for the sur-face air temperature tendency (at the level p0 2 Dp/4) is

]Tsa ˙ ˙5 2J(c , T ) 1 Q 1 Q ,1/2 sa 1 2]t

where J(c1/2, Tsa) is calculated using the streamfunctionc1/2 5 (c0 1 c1)/2, Q1 is the thermal diffusion, and Q2

is the radiative cooling. Here Dp is equal to 1000 hPa/(N 2 1), where N is the number of vertical levels atwhich temperature is defined in the model.

c. Ekman momentum approximation

Bannon and Salem (1995) used a multiple-scale anal-ysis of the baroclinic Ekman layer to show that themomentum equations used in this model are accurateto O(E). Here E is the Ekman number equal to 2n/( fD2),where D is the vertical scale of the flow. These equationswill be referred to as the Ekman momentum approxi-mation.

The ageostrophic winds are calculated from the sur-face shear stress by assuming that the first-order balancein the momentum equation is between the Coriolis forceand the vertical diffusion of momentum,

](v 1 v )g a2n r g 5 2c r (v 1 v ).M d s g a]p

The drag coefficient is equal to the surface drag timesthe ambient wind, cd 5 csurface|vs |, and is chosen to beequal to 3 3 1022 m s21. The ageostrophic winds areassumed to go to zero above 700 hPa, but in fact thevalues are only on the order of the geostrophic windsat the first model level and at the surface.

d. Diabatic heating

Diabatic heating is parameterized in this model by arelaxation back to a radiative–convective equilibriumstate with a constant diabatic heating timescale. Thediabatic heating timescale in the standard model is setequal to 40 days.

The radiative–convective equilibrium state is repre-sented by a horizontally averaged mean lapse rate,dTe/dz, plus horizontal temperature gradients, dTe/dy.The lapse rate in radiative–convective equilibrium ischosen to be a constant 27.08 km21 in the troposphere.The lapse rate is set equal to zero above the jet maxi-mum. The radiative equilibrium lapse rate would bemuch larger, closer to 2158 km21 in the troposphere.Since all moist processes have been neglected in thismodel, the modification of the radiative equilibrium pro-file by moist convection is taken into account implicitlyby assuming that moisture brings the lapse rate close toa moist adiabat.

The latitudinal temperature distribution is chosensuch that the potential temperature gradients are con-stant with height in the troposphere,

u(y, p) 5 221.5 sin[p(y 2 L/2)/(L/2)],

for L/4 # y # 3L/4, where L is the total channel width—which is set equal to 10 000 km in the standard run—and u is potential temperature. This gives a temperaturecontrast of 308 over the center 2500 km of the channel,compared to an estimate of 348 for the wintertime tem-perature gradients in the lower troposphere from Man-abe and Moller (1961). A region with zero temperaturegradients is extended 2500 km to the north and southof the sine function to ensure that the channel walls donot modify the dynamics. The gradients in the strato-sphere are positive, constant with height, and 10 timessmaller than the gradients at 375 hPa.

e. Turbulent heat fluxes in the ABL

The lower boundary is assumed to be a sea surfacewith fixed temperature, usea. A linearized bulk aerody-namic drag formula is used to calculate the surface heatflux,

FSH 5 2cdcprs(uair 2 usea),

where rs is the surface density. The surface air tem-perature uair is assumed to be equal to the potential tem-perature at the first level, which in the standard 17-levelmodel is 31 hPa over the surface. The drag coefficient,cd, is chosen to be equal to 3 3 1022 m s21, althoughin general, the drag coefficient would be a function ofthe stability.

A diffusive parameterization of turbulent heat fluxesin the ABL are included in the model. Without thesefluxes the static stability at the lowest level becomesvery large in the equilibrated state because the eddiestransport heat vertically, causing the lowest levels to becold relative to observations. These fluxes are param-eterized as

3]u p

2 2 21F 5 n (p)r gc n (p) 5 5 m s ,SH T p T 1 2]p p0

where p0 is the pressure at the lower boundary. Theheating due to the vertical diffusion of heat is then cal-culated from the fluxes as

]u g ]FSH5 .]t c ]pp

f. Friction in the ABL

Friction is parameterized by a linearized bulk aero-dynamic drag at the surface and diffusive mixing abovethe surface, in the same way as was outlined for thediabatic heating.

The linearized bulk aerodynamic drag formula is usedto calculate the wind stress at the surface,


FIG. 1. Vertically and time-averaged eddy kinetic energy, KEP (onthe left), and eddy enstrophy spectra, EN (on the right), vs totalwavenumber, K, for the horizontal resolution and domain runs (SR,L/2, L/4, DX/2). The vertical and horizontal scales are logarithmic.For comparison lines with slopes of 23 and 21 are shown for theKEP and EN spectra, respectively. (a) and (b) show the results forthe standard run; (c) and (d) for one-half the standard domain size;(e) and (f ) for one-quarter the standard domain size; and (g) and (h)for twice the standard horizontal resolution.

t s 5 2cdrs(vg 1 va).

The surface wind is calculated assuming that the tem-perature at p 5 ps is in thermal wind balance with thewinds at p 5 p1 and p 5 p0. The drag coefficient, cd,in the standard run is again set equal to 3 3 1022 ms21.

Above the surface, the turbulent shear stress is cal-culated using a flux-gradient formulation,

](v 1 v )g a2t 5 r[v*v*] 5 n (p)r g ,M M ]p

where

3p2 21n (p) 5 5 m s .M 1 2p0

The vertical diffusion of momentum by the turbulenteddies is then calculated from the shear stress as F 5g]tM/]p.

3. Resolution and domain size dependence of theequilibrated states

In order to determine if the dynamics of the modelare convergent at 17 vertical levels, the model has beenrun using the same parameter values but with twice thevertical resolution. Since it has been found in other stud-ies that the wave that is responsible for the most energytransport may be the largest wave resolved by the model(Cehelsky and Tung 1991), the model has also been runwith half the channel length and a quarter of the channellength of the standard run, using the same resolution asthe standard run. Finally, to check the horizontal res-olution of the standard run, it was rerun with doublethe resolution of the standard run, both meridionallyand zonally.

a. Horizontal resolution and domain size

Figure 1 shows how the vertically averaged kineticenergy and enstrophy spectra depend on horizontal res-olution and domain size. The wavelength of the wavethat is responsible for the majority of the potential vor-ticity and heat transport is approximately equal to 4000km for all these runs (results not shown). Thus this waveis not saturated in our standard run (cf. Cehelsky andTung 1991 and Welch and Tung 1998). Figure 1 dem-onstrates that it is necessary to use a channel lengthlarger than the dominant heat transporting wave to re-solve the turbulent cascades, and that the domain sizeand horizontal resolution of the standard run are ade-quate to resolve the turbulent cascade.

The temperature structure and zonal wind structureare insensitive to the changes in horizontal resolutionand domain size (results not shown). The potential vor-ticity gradients in the equilibrated state are also insen-sitive to the changes in horizontal resolution and domain

size below 600 hPa. However, in the upper tropospherethe potential vorticity gradients are more sensitive andcan differ by up to b.

b. Vertical resolution

An additional run with 33 vertical levels demonstrat-ed that the potential vorticity gradients in the ABL aredependent upon the vertical resolution. This is becauseit is the mixing of potential temperature and not themixing of potential vorticity that is the primary dynamicwithin the ABL. Therefore, the potential vorticity gra-dients will always be a function of resolution. The po-tential temperature structure on the other hand is insen-sitive to the increase in resolution (results not shown).

In previous studies of the impact of variable verticalresolution on the Boussinesq dynamics of a midlatitudeb-plane channel model it has been concluded that it isnecessary to use at least 10 to 20 levels to resolve theeddy dynamics (Pavan 1996; Hua and Haidvogel 1986).


Our results are consistent with this conclusion. Pavan(1996) also concluded that only 3 levels are necessaryto reproduce the general features of the response. Themodel used by Pavan (1996), however, did not attemptto include thermal diffusion (which dominates the dia-batic heating in the ABL) or the feedbacks that maintainthe static stability. Lower vertical resolution runs thatwe carried out showed that including these processescause the model to be more resolution dependent (resultsnot shown).

4. Equilibrated model climate compared toobservations

The standard model run has been chosen to approx-imate observations of the Northern Hemisphere winterclimate as closely as possible, given the simple repre-sentations of heating and friction and the constructionof the ABL in the model.

a. Distributions of pseudo–potential vorticity

Figure 2a displays the January 1982–94 zonal meanmeridional gradient of Ertel’s potential vorticity (EPV)divided by the contribution from the planetary vorticity,for the Northern Hemisphere January 1982–94 clima-tology, produced from the NCEP–NCAR reanalysis da-taset. Figure 2a shows that the EPV in midlatitudes ischaracterized by large gradients below 800 hPa and atthe tropopause. Below 800 mb, the meridional temper-ature gradients are large due to the heating of the at-mosphere by the underlying surface. The heating of theair near the surface causes turbulent fluxes of heat andmomentum to mix momentum and heat, resulting insmall values of static stability near the ground. LargeEPV gradients are found at the tropopause where thehigh EPV stratospheric air meets the low EPV tropo-spheric air.

The region of EPV homogenization can be seen clear-ly in Fig. 2a where the EPV gradients between 600 and750 hPa are less than b. This is the region where thelarge-scale baroclinically unstable Rossby waves havea critical level. Irreversible mixing of EPV along theisentropes reduces the EPV gradients and the slope ofthe isentropes in this region. The levels between thisregion and the tropopause have reduced EPV gradientsrelative to radiative–convective equilibrium estimatesbut they are still large relative to b.

Figure 2b displays the potential vorticity gradients,divided by b, for the equilibrated state of the standardrun. Comparing the observations, Fig. 2a, to the equil-ibrated state of the model, Fig. 2b, we see that the qual-itative features of the observed potential vorticity dis-tribution are resolved by the model. Quantitatively thepotential vorticity gradients in the ABL are larger thanthe observations.

b. Structure of the mean temperature

Figure 3a compares the static stability of the standardrun to observations. Since the eddy diffusion coefficientis a continuous function of height, the large potentialvorticity gradients in the ABL extend higher, and thestatic stability peaks at a higher level than in the ob-servations. The static stability in the standard run below700 hPa is significantly smaller than in the observations.The magnitude and vertical profile of the static stabilityabove 700 hPa approximate observations.

Model runs that have diffusion coefficients that go tozero below 750 hPa have a much more realistic meanstatic stability and a larger region of potential vorticityhomogenization. The continuous eddy diffusion coef-ficient has been chosen for the standard run to allowthe region over which the vertical eddy heat fluxes playa significant role to depend on the other parameters inthe model. Limiting the thermal diffusion to a regioncloser to the surface would yield a more realistic rep-resentation of a thermal capping inversion.

Figure 3b compares the meridional temperature gra-dients of the standard run to observations. The largethermal damping in the ABL prevents very much re-duction of the meridional temperature gradients at thelowest levels by the large-scale eddies. These averagedgradients are seen to be too large relative to the averagedgradients in observations.

Above the ABL, the potential vorticity has been ho-mogenized (Fig. 2b). This homogenization extendsalong the sides of the jet, where the large-scale waveshave a critical level. The homogenization is a result ofboth increased static stability and decreased horizontaltemperature gradients. This region of homogenizationapproximates the homogenization of EPV that is seenin observations (Figs. 2a and 2b).

Above the homogenized potential vorticity region, thepotential vorticity gradients increase with height andhave a maximum at the tropopause (Fig. 2b). In theupper troposphere, the meridional temperature gradientsfrom observations are significantly less than those ofthe standard run. The dynamics that are responsible forthe observed temperature gradients in the upper tro-posphere are not adequately resolved in this simple pro-cess model.

c. Structure of the zonal mean eddy fluxes

The model parameters have been chosen so that thezonal mean eddy sensible heat fluxes of the standardrun approximate observations of the Northern Hemi-sphere wintertime eddy sensible heat fluxes. Therefore,the zonal mean eddy sensible heat fluxes from the modelare compared to the zonal mean stationary plus transienteddy sensible heat fluxes of the observations. Sensibleheat fluxes will be referred to as heat fluxes in whatfollows.

Figure 4b shows the zonal mean meridional eddy heat


FIG. 2. (a) Zonal mean meridional gradient of potential vorticity (PV) [2g( f 1 z)(]p/]u)21]on isentropes, calculated from zonal mean u, divided by the b contribution to the potential vorticity[2gb(]p/]u)21], for Jan 1982–94. Data from the NCEP–NCAR reanalysis project (Kalnay et al.1996). (b) Zonal mean meridional gradient of pseudo–potential vorticity divided by b from theequilibrated state of the standard run. Contours of 60.5 b have been added to indicate the regionsof PV homogenization.

fluxes of the equilibrated standard run and observations.Figure 4a shows the zonal mean vertical eddy heat flux-es, of the equilibrated standard run and observations.The magnitude and vertical structure of the model fluxesresemble the observed fluxes qualitatively. Both the ver-tical and meridional eddy heat fluxes peak at 875 hPa,due to the thermal damping in the ABL. Without thethermal damping the fluxes peak at the surface. Themajor inadequacy of the model fluxes is the small sec-ondary peak in the meridional eddy heat flux in theupper troposphere. Since the dynamics of this modelare dry, the energy transport due to water vapor fluxeshas been neglected. These fluxes are responsible forapproximately 35% of the total energy transport in the

lower troposphere (Peixoto and Oort 1992), and theirabsence contributes to the model’s overestimate of themeridional temperature gradients in the lower tropo-sphere.

Figure 4c shows the zonal mean meridional circula-tion heat fluxes of the equilibrated standard run andobservations. The potential temperature transport hasbeen calculated instead of the temperature transport totake into account the cancellation between the sensibleheat flux and the potential energy flux. This cancellationis only significant in the mean fluxes since the eddypotential energy flux is small relative to the eddy sen-sible heat flux (Peixoto and Oort 1992). Given the dif-ference between the two observational datasets, the


FIG. 3. Comparison of (a) static stability and (b) meridional tem-perature gradients of the equilibrated state of the standard run withNCEP–NCAR Jan 1982–94 observations.

FIG. 4. Comparison of (a) eddy vertical heat fluxes (b) eddy me-ridional heat fluxes (c) mean meridional heat fluxes (d) eddy mo-mentum fluxes of the equilibrated state of the standard run (3) withJan NCEP–NCAR (V) 1982–94 and ECMWF (1) 1985–89 datasets.

magnitude and vertical distribution of the model’s meanmeridional fluxes closely approximate the observed dis-tribution. The vertically integrated mean meridional heatflux in the standard run is much smaller than the ver-tically integrated eddy heat flux (23 K m s21 vs 116K m s21), just as in the observations.

Figure 4d shows the zonal mean eddy momentumfluxes for the standard run compared to observations.This figure shows that the model results resemble thevertical structure and magnitude of the observationsquite well, except at the surface. At the lowest level,the observations have negative momentum fluxes, whilethe model has positive fluxes.

5. Time evolution to the equilibrated basic state

Figure 5 shows the time evolution of the zonal meanpotential vorticity gradients at the center of the channelfor the first 60 days of the integration of the standardrun. At day 0, the model is in approximate RCE and aneddy perturbation is introduced into the model. Duringthe first 12 days the eddy effects are small, and a sym-metric state continues to evolve slowly. Then exponen-tially growing eddies reach large enough amplitudes thattheir fluxes start to modify the mean potential vorticitygradients. The modification of the mean flow starts in

the ABL with an increase in the magnitude of the po-tential vorticity gradients at the surface and a decreaseat the top of the ABL, approximately 700 hPa. Thisadjustment is due to the vertical eddy heat transportsthat cool the surface and heat the upper region of theABL. The efficient mixing of potential temperature dueto fluxes across the air–sea interface heats the lower partof the ABL causing the static stability to become verysmall there. This decrease in the static stability increasesthe slope of the isentropes and increases the potentialvorticity gradients. In the upper region of the ABL thevertical eddy fluxes of heat reduce the slope of the is-entropes and decrease the potential vorticity gradients.The wave–mean flow adjustment propagates upward asthe eddies become deeper. The flow becomes approxi-mately steady at day 30.

The amplitude of the perturbation potential vorticitybecomes steady when the potential vorticity mixingreaches a maximum in the midtroposphere. The equil-ibrated perturbation potential vorticity has a primarypeak at the surface and a secondary peak at the tro-popause (figure not shown). There is a minimum in thelower troposphere where the potential vorticity gradi-ents have been homogenized. The flow becomes ap-proximately steady at day 30. The evolution is consistentwith the picture suggested by Stone and Nemet (1996)—


FIG. 5. Time evolution of the zonal mean potential vorticity gradients at the center ofthe channel from the RCE state to the equilibrated state from the standard run, in unitsof b.

that is, the baroclinic eddies effectively homogenize thepotential vorticity field in midlatitudes in the lower tro-posphere—but this outcome is prevented in the ABL bydiffusion and dissipation, and the upper troposphere re-mains near radiative–convective equilibrium.

6. Maintenance of the perturbation potentialvorticity in the equilibrated state

The balance of terms in the perturbation potentialenstrophy tendency equation clearly shows the eddyfeedbacks that are maintaining the wave activity. Thisequation is derived by multiplying the equation for theperturbation potential vorticity tendency by the pertur-bation potential vorticity. The equation is

2 21 ]q9 ]q 1 ]y9q95 y9q9 2

2 ]t ]y 2 ]y

˙] g ] ]u QR1 q9 2 f rn 1 1 k ·= 3 F ,0 T5 1 2 6]p p ]P ]z spCp

where y is the meridional wind, p is the Exner function,primes designate deviations from the zonal mean, anda bar designates a zonal mean.

Perturbation potential enstrophy is equivalent to thesquared magnitude of the potential vorticity anomalies.The terms in the perturbation potential enstrophy ten-dency equation indicate which processes are responsiblefor the maintenance of the anomalies. The first term onthe right-hand side is the quasi-linear flux of potential

vorticity and will be referred to as the downgradienttransport of perturbation potential vorticity because thecorrelation between the mean potential vorticity flux andthe mean potential vorticity gradients tends to be neg-ative. Therefore, this term can be thought of as thedowngradient transport of perturbation potential vortic-ity that reduces the mean potential vorticity gradientsand is a source of perturbation potential vorticity. Thesecond term on the right-hand side includes the flux ofperturbation potential vorticity by wave–wave interac-tions. The last large expression on the right-hand sideconsists of the vertical diffusion of heat, Newtoniancooling, and the vertical diffusion of momentum, re-spectively. All of these act to damp out the perturbationpotential vorticity, except for the thermal diffusion atthe second model level due to the large decrease in thestatic stability.

A plot of the terms in the perturbation potential en-strophy tendency equation for the standard run is shownin Fig. 6. This figure shows that the nonlinear wave–wave interactions become significant above the bound-ary layer. The balance of terms in the perturbation po-tential enstrophy equation shows that the nonlinear cas-cade of perturbation potential vorticity and the dampingterms contribute equally to balancing the quasi-linearpotential vorticity transport between 500 and 600 hPa.Within the ABL the dynamical balance is between thequasi-linear potential vorticity transport and the forcing(which is dominated by the thermal diffusion). The non-linear wave–wave cascade is insignificant within the


FIG. 6. Terms in the perturbation potential enstrophy tendencyequation for the standard run for (a) above the ABL and (b) withinthe ABL.

ABL, and the dynamics are essentially thermally dis-sipating Rossby waves as discussed by McIntyre andNorton (1990).

Figure 7 shows the time evolution of the zonal meanpotential vorticity gradients above the ABL from theRCE state to the equilibrated state. Figure 7 shows thatthe zonal mean potential vorticity gradients above theABL become homogenized within 5 days of when theeddies become finite amplitude (approximately day 15).Around day 30 the potential vorticity gradients at thetropopause become quasi-steady (results not shown).The jet fluctuates about the channel center and has po-tential vorticity gradients that range between 4 and 6 b.

Figure 8 shows the time evolution of the amplitudeof the perturbation potential vorticity for waves 1–10from the standard run. This figure shows that wave 6dominates in the initial phase of the evolution of theeddy regime, and that its initial growth is consistentwith standard models for the most unstable wave ofrealistic symmetric states. However, in the equilibratededdy regime, waves 4 and 5 dominate, with wave 4being somewhat larger.

The adjustment from the symmetric state to the equil-ibrated state occurs over a very short period, approxi-mately 5 days. Therefore, if the external forcing changes

on timescales of a week or longer, the eddies will havetime to adjust the equilibrated state through wave–meanflow adjustment.

7. Nonlinear versus quasi-linear equilibrations

In order to isolate the importance of resolving thenonlinear cascade, a separate experiment was carriedout, with the static stability held fixed at the equilibratedstatic stability of the standard run, with only advectiveterms that modify the zonal mean flow retained in thetendency equations, and with only wave numbers 4 to6 retained. We refer to this experiment as the quasi-linear run. Therefore in this run, waves 4, 5, and 6 candirectly modify the mean flow but can only modifythemselves indirectly through changes in the zonal meanflow. The initial state of the quasi-linear run has me-ridional temperature gradients equal to the radiative–convective equilibrium meridional temperature gradi-ents of the standard run and static stability equal to theequilibrated static stability of the standard run.

Figure 9 shows the equilibrated zonal mean potentialvorticity gradients from the quasi-linear run (cf. theequilibrated zonal mean potential vorticity gradientsfrom the standard run Fig. 2b). It is seen that the ho-mogenization of potential vorticity above the boundarylayer in the quasi-linear run is simulated without thenonlinear cascade. This is because most of the adjust-ment above the ABL is due to the increase in the staticstability there (figure not shown), and very little ad-justment by the meridional heat fluxes is needed to com-pletely eliminate the potential vorticity gradients abovethe ABL. The main difference between zonally averagedpotential vorticity gradients of the two runs is that thehomogenized region is broader in SR.

A Fourier decomposition of the time evolution of theperturbation potential vorticity shows that the quasi-linear run is essentially maintained by wave 5, since thewave 4 perturbation potential vorticity decays away (fig-ure not shown) and wave 6 is always small. Thus giventhe static stability, wave 4 in SR has to be maintainedby the upscale energy cascade. This contrasts with thetwo-layer model results of Whitaker and Barcilon(1995), who found that the dominant wave is maintainedby baroclinic generation.

Two further runs were carried out to determine howdifferent the equilibrated state would be if the mainte-nance of the equilibrated state was strictly due to eitherwave 4 or wave 5; that is, these two runs retain onlywave 4 or wave 5, respectively. They started from thesame initial condition as the quasi-linear run. Compar-ing these runs and the quasi-linear run to SR it is pos-sible to determine whether the fully nonlinear run equil-ibrates to a state determined mainly by wave 4, wave5, or possibly something completely different. Figure10 shows the difference between the potential vorticitygradients in SR and in the wave 4 only and wave 5 onlyruns, at the center of the channel. Also shown is the


FIG. 7. Time evolution of the zonal mean potential vorticity gradients at 718.75 hPa from theRCE state to the equilibrated state from the standard run, in units of b.

FIG. 8. Time evolution of the amplitude of the perturbation potentialvorticity for wavenumbers 1–10 from the standard run. ‘‘All-456’’shows the amplitude contained in all wavenumbers other than 4, 5,and 6.

difference in this quantity between SR and its initialvalue in the wave 4 or 5 only runs. Thus comparing thethree curves in Fig. 10 shows us how much the potentialvorticity gradients in the wave 4 and wave 5 only runshave changed from their initial conditions, and howmuch they differ from SR. We see that the wave 4 onlyrun hardly modifies the mean flow, and is quite far fromthe SR result. In addition we see that the structure of

the potential vorticity gradients above the ABL in SRis closely equal to the equilibrated potential vorticitygradients in the wave 5 only run. Within the ABL theequilibrated potential vorticity gradients from SR aresomewhere between the equilibrated potential vorticitygradients due to wave 4 or wave 5 only.

A comparison of the time evolution of the meridionaleddy heat fluxes from SR and the wave 5 only run at875 and 250 hPa, the primary and secondary eddy heatflux maxima, shows that the magnitudes are equivalentbut that the wave 5 only run has much larger and moreregular oscillations at 875 hPa (figure not shown). Thisdemonstrates that the mean fluxes due to wave 5 ap-proximate those from the fully nonlinear run. The wave4 only run was found to have fluxes that were approx-imately 2 times smaller than SR (figure not shown). Inaddition, a Fourier decomposition of the SR run (notshown) found that both the vertical and meridional eddyheat fluxes in SR are dominated by wave 5. Thus weconclude that the equilibrated state in SR is maintainedprimarily by the interaction of wave 5 with the zonalmean flow.

8. Stability characteristics of the equilibrated state

This section looks at the role of instabilities in themaintenance of the equilibrated state. First we describesome more properties of the equilibrated state. Changesin the total domain-averaged eddy kinetic energy remainsmall in the equilibrated state even though the model is


FIG. 9. Latitude–pressure cross section of the zonally averaged potential vorticity gradients inthe equilibrated state of the quasi-linear run, in units of b.

FIG. 11. Time evolution of the eddy kinetic energy (KEP), thezonal mean wind (UM), and the meridional gradient of potentialvorticity (QY) at 281 hPa at the center of the channel from thestandard run for days 1–130. Scaling is described in the text.

FIG. 10. Comparison of the potential vorticity gradients at the centerof the channel in the equilibrated state of the standard run with thequasi-linear runs, in units of b.

clearly not steady (figure not shown). Figure 11 showsa comparison of the eddy kinetic energy (KEP), thezonal mean wind (UM), and the mean potential vorticitygradients (QY) for the first 130 days of the standard runat 281 hPa. The UM and QY are normalized by theirinitial values, 41.3 m s21 and 8.2 b, respectively, whileKEP is normalized by the maximum value during thisperiod, 444 m2 s22, so that the fields can be easily com-

pared. This figure shows that at the tropopause there isa clear negative correlation between QY and KEP. Themaximum correlation is at a zero time lag. The meanpotential vorticity gradients are a minimum (maximum)when the eddy kinetic energy is a maximum (minimum).Therefore, the changes to the zonal mean state at the


FIG. 12. Domain-averaged time evolution of the total KEP andKEP per wavenumber at the center of the channel from the standardrun for days 200–400. In units of m2 s22.

tropopause occur out of phase with the changes in theeddy kinetic energy. There is a clear positive correlationbetween the magnitude of the zonal mean wind and themean potential vorticity gradients at the tropopause. Themagnitude of the zonal mean wind at the tropopause,but not the meridional structure of the zonal mean wind,is highly correlated with the time evolution of the mag-nitude of the potential vorticity gradients at the jet. Thisresult suggests that the barotropic governor proposedby James (1987) is not an important mechanism for thestabilization of the waves during this time, in agreementwith the baroclinic turbulence study of Welch and Tung(1998).

Figure 12 shows the total domain-averaged KEP andthe KEP contributed to the domain average by eachwavenumber. Figure 12 clearly shows that waves 4 and5 are responsible for the majority of the KEP and thatthey are highly anticorrelated. All other waves contrib-ute very little to the total domain-averaged KEP. Com-paring the total KEP and its growth rates to those cal-culated for wave 4 alone, it is clear that large changesin KEP due to waves 4 and 5 cancel keeping the changesin the total KEP small. The correlation coefficients atzero time lag between the growth rate of the total KEPand the growth rate of waves 4 and 5 are 0.19 and 0.22,respectively. The correlation coefficient at zero time lagbetween the growth rate of total KEP and waves 4 and5 together is 0.74. Therefore, the joint response of waves4 and 5 plays an important role in the variance of KEPin SR.

In order to get a sense of whether or not instability isplaying a role in the maintenance of the equilibrated meanstate, three-dimensional linear stability analyses(3DLSAs) were carried out. All of the forcings used inthe fully nonlinear model are linear and are thereforeincluded in the 3DLSAs. The full model is linearized

about various basic states calculated from instantaneousbasic states from the standard run following Jin and Hos-kins (1995). The static stability is held fixed at the valuescalculated from the composite basic states. Compositeswere made for the period between days 200 and 400.One 3DLSA was carried out that linearized the modelabout the two-dimensional zonal mean time mean statewithout waves. This composite was found to be linearlystable with the least damped mode being wave 6.

An additional 3DLSA was then run with the basicstate taken to be the same two-dimensional zonal meanstate plus wave 4. The phase speed of the wave wascalculated by averaging the phase speed of the wavebetween days 200 and 400 from SR. The average phasespeed for wave 4 was 2.5 m s21. The average phasespeed was found to be the same at 720 and 218 hPaand showed little variance throughout the integration(figure not shown). The stability analysis was carriedout in the reference frame moving with the wave in-cluded in the basic state (Lin 1980). Although this stateis, strictly speaking, not a solution of the time meanequations, we carried out this 3DLSA in an attempt togain insight as to how wave 5 might be generated inSR. This state was also found to be linearly stable. Theleast-damped normal mode was dominated by wave 5.

Finally we carried out a third 3DLSA, in which thebasic state is a composite of the instantaneous stateswhen wave 4 was a maximum in SR. Again, this basicstate is not strictly a solution of the mean equations,but in this case it is at least a state with zero initialtendency, so that the 3DLSA should show whether thereis any initial normal-mode growth. In this case therewas. The mode was dominated by wave 6, and had agrowth rate of 0.17 day21. This mode represents aninstability of wave 4, since when this 3DLSA was re-peated with wave 4 removed from the basic state, it wasno longer unstable. This instability is one likely sourceof wave 6, which is the third-largest wave present inSR.

These results suggest the following scenario for theprocesses that control the growth and decay of the wavesin the model. When wave 4 is a minimum the flow isstable and the energy cascades upscale, particularly fromwave 5 to wave 4, so that wave 4 grows and wave 5decays. Eventually, wave 4 becomes large enough sothat the flow becomes unstable and the growth of thewaves is dominated by wave 6. This wave quickly sat-urates (see Fig. 12), and gives up its energy to largerscales. The growth of wave 5 while wave 4 is decaying(see Fig. 12) suggests that in effect the energy that wave6 extracts from wave 4 by linear instability is then sup-plied to wave 5 by nonlinear wave–wave interactions.Once wave 4 has become small enough, the flow be-comes stable, wave 5 now loses energy to wave 4, sowave 5 decays and wave 4 grows, and the cycle repeatsitself. The interaction between wave 4 and 5 in SR sup-presses the much larger oscillations in KEP seen in thewave 5 only run described in section 6, but in both cases


the mean properties of wave 5 and its interaction withthe mean flow are very similar, so the mean equilibratedstates are similar in both runs.

9. Discussion

The results of this study suggest an equilibrated statethat is in many ways similar to the midlatitude tropo-sphere. This equilibrated state is maintained by quasi-linear dynamics and can be thought of as a quasi-neutralstate. This neutral state can best be understood in termsof a coupling between the efficient vertical mixing ofpotential temperature in the ABL with the efficient me-ridional mixing of potential vorticity in the lower tro-posphere. This is very significant in two aspects. First,changes in the surface temperature will be reflected inthe potential temperature at the top of the ABL, whichprovides a boundary condition for the mixing of poten-tial vorticity in the free troposphere. Therefore, changesin the surface temperature may significantly impact thetemperature structure of the free troposphere. Second,this study suggest that the efficient homogenization ofpotential vorticity near the steering level may be a robustfeature of the equilibrated state, that is, insensitive tochanges in forcing. Both of these aspects will be ex-plored in two companion papers that look at the sen-sitivity of the model’s equilibrated state to changes inforcing and dissipation.

Our study suggests a novel scenario of equilibrationdue to wave–mean flow interaction. The total growthrate of the eddy kinetic energy was found to be essen-tially determined by the feedback between wave 5,which was responsible for the majority of the wave–mean flow fluxes, and wave 4, which was the wave withthe largest amplitude and depends on the upscale energycascade for its existence. The equilibrated state’s zonalmean state was found to be linearly stable, but it waslinearly unstable part of the time when a sufficientlylarge wave 4 component was present and was added toits zonal mean basic state. The most rapidly growingmode in the linear stability analysis about the wavy basicstate was found to be dominated by a wave 6. From ourresults it appears that the variance in the fully nonlinearrun was maintained in part by the feedback between theenergy cascade to the larger scales and the extractionof energy from the larger-scale waves by shorter linearlyunstable waves. Linearizing about the zonal mean basicstate overestimates the stability of the equilibrated state,and excludes this significant mechanism. The wave–wave fluxes appear to play a different role than in theWelch and Tung (1998) saturation scenario. First, wave5 is responsible for the majority of the wave–mean flowfluxes in part because shorter waves saturate, but alsoin part because it can draw energy from wave 4 withthe aid of smaller-scale instabilities. Second, eventhough there is a nonlinear cascade and an energy trans-fer to larger scales, the equilibrated state of the fullynonlinear run was shown to closely approximate the

neutral state of the wave 5 only run. The nonlinearwave–wave fluxes were shown to play a significant rolein the equilibration by increasing the amplitude of wave4. This in turn appears to cause a mode dominated bywave 6 to become linearly unstable.

The episodic growth of waves 4 and 5 in the equil-ibrated state appears to be consistent with Farrell’s(1985) picture of baroclinic eddy growth associated withnonnormal modes. Wave–wave interactions are essentialin giving rise to these waves; nevertheless both drawon the mean available potential energy and transportheat poleward, and we could find no normal modes thatcould by themselves account for their growth. However,the equilibrated state also contains eddies (primarilywave 6) that appear to grow because of normal-modeinstability, and these appear to facilitate the exchangeof energy between waves 4 and 5. Thus the equilibrationinvolves a mixture of Farrell’s mechanism and the clas-sic mechanism of normal-mode instability, although, tobe sure, it is a wave instability rather than an instabilityof the zonal mean flow. This more complicated pictureis likely to be sensitive to the particular parameter choic-es we have made, and we plan to investigate its depen-dence on the parameters in subsequent papers.

The results of this study are consistent with the resultsof Simmons and Hoskins (1978) where the reduction oflower-tropospheric temperature gradients due to baro-clinic growth preceded the modification of the jet struc-ture due to barotropic decay in the life cycle of a non-linear baroclinic eddy. This study extends the studies ofeddy life cycles to a fully equilibrated state that is forcedby both heating and friction, and in which eddies arealways present. The results of this study suggest anequilibrated state that may be relevant to dynamics inthe atmosphere.

The equilibrated state in our model has a verticalstructure consistent with the suggestion of Stone andNemet (1996) that the vertical structure of the potentialvorticity (PV) gradients in the midlatitude tropospherehas distinct regimes: a region of large PV gradientswithin the ABL and a region of approximately homo-genized PV between 600 and 750 hPa. The results ofthis model study have demonstrated how these regimescome about in one particular set of circumstances. Howthese results depend on the forcing and dissipation willalso be explored in the companion papers referred toearlier. However, they do show that the results of thetwo-layer models, which cannot resolve these differentregimes, need to be treated with caution.

Acknowledgments. This work was supported in partby the Global Atmospheric Modeling and Analysis Pro-gram of the National Aeronautics and Space Adminis-tration, directed by Ken Bergman, under Grant NAG5-4880 to MIT. We are indebted to Richard Lindzen formany useful discussions. The International Pacific Re-search Center is partly sponsored by Frontier ResearchSystem for Global Change.


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