Comprehensive meteorological modelling of the middle atmosphere: a tutorial review

Journal of Atmospheric and Terresrrial Physics, Vol. 58, No. 14, pp. 1591-1627, 1996 Copyright 0 1996 Elsevier Science Ltd Pergamon

PII: SOO21-9169(%)00028-l Printed in Great Britain. All rights reserved

0021-9169/96 $15.CO+O.O0

Comprehensive meteorological modelling of the middle atmosphere: a tutorial review*

Kevin Hamilton

Geophysil;al Fluid Dynamics Laboratory/NOAA, Princeton University, P.O. Box 308, Princeton, NJ 08542, U.S.A.

(Received 6 December 1995 ; accepted 30 January 1996)

Abstract-This paper reviews the current state of comprehensive, three-dimensional, time-dependent modelling of l:he circulation in the middle and upper atmosphere from a meteorologist’s perspective. The paper begins with a consideration of the various components of a comprehensive model (or general circulation model, GCM), including treatments of processes that can be explicitly resolved and those that occur on scales too small to resolve (and that must be parameterized). The typical performance of GCMs in simulating the tropospheric climate is discussed. Then some important background on current ideas concerning the general circulation of the stratosphere and mesosphere is presented. In particular, the transformed-Eulerian mean flow formalism, the role of vertically-propagating internal gravity waves in driving the large-scale circulation, and the notion of a stratospheric surf zone are all briefly reviewed. Using this background as a guide, some middle atmospheric GCM results are discussed, with a focus on simulations made recently with the GFDL ‘SKYHI’ troposphere-stratosphere-mesosphere GCM. The presentation attempts to emphasize the interaction between theory and comprehensive modelling. Many theoretical notions cannot be confirmed in detail from observations of the real atmosphere due to the various limitations in the observational methods, but can be very completely examined in GCMs in which every atmospheric variable is known perfectly (within the limits of the numerical methods). It will be shown that our understanding of both the role of gravity waves in the general circulation and the nature of the stratospheric surf zone has benefited from analysis of GCM results.

From the point of view of the upper atmosphere, one of the most interesting aspects of GCMs is their ability to generate a self-consistent field of upward-propagating gravity waves. This paper concludes with a discussion of the gravity wave field in the middle atmosphere of GCMs. Comparisons of the explicitly- resolved gravity wave field in the SKYHI model with observations are quite encouraging, and it seems that the model is c.lpable of producing a gravity wave field with many realistic features. However, the simulated horizontal spectrum of the eddy momentum fluxes associated with the waves is quite shallow, suggesting that much of the spectrum that is important for maintaining the mean circulation is not explicitly resolvable in current GCMs. A brief discussion of current efforts at parameterizing the mean flow effects of the unresolvable gravity waves is presented. Copyright 0 1996 Elsevier Science Ltd

1. INTRODUCTION

This paper will pre:sent an informal survey of some current issues in the area of comprehensive modelling of the middle and upper atmospheric circulation. It will try to give a meteorologist’s perspective on atmospheric modelling and will assume that the reader has a general familiarity with fluid mechanics as applied to the atmosphere, and a basic knowledge of those aspects of atmospheric dynamics that traditionally have been of particular interest to aeronomers (gravity waves, tides etc.). The readers will be provided with enough background. that they can understand some

* Based on a Tutorial Lecture Presented at the 10th Annual CEDAR Workshop in Boulder Colorado on June 29th, 1995.

of the current issues in the dynamics in the middle atmosphere and can gain some perspective on the contribution comprehensive models have made and may be able to make to these issues. Section 2 begins with a consideration of the problems that need to be addressed in actually formulating a comprehensive meteorological model. In Section 3 a general idea of how successful such models are in simulating the tropospheric circulation is given. Section 4 is a review of some basic concepts concerning our understanding of the stratospheric general circulation. This will serve as background to Section 5 in which the performance of current models in the stratosphere and mesosphere is discussed. In Section 6 aspects of model simulations of the vertically-propagating gravity wave field, perhaps the most important contribution comprehensive

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models can make to our understanding of the dynamics of the mesosphere and lower thermosphere are considered.

First let me define what I mean by a ‘comprehensive’ model, or what meteorologists refer to as a ‘general circulation model’ (GCM). The term GCM has been coined to describe three-dimensional, time-dependent, simulation models of the global atmosphere that employ a complete nonlinear version of the Navier- Stokes equations (typically simplified by a hydrostatic assumption) and that include both a sophisticated treatment of the radiative transfer problem and an attempt to explicitly include the hydrological cycle. This definition can encompass a wide range of models with different levels of sophistication. In particular, the treatment of the lower boundary can range from a simple prescription of the ocean surface temperatures and an assumption of zero net energy input to the land surface, to models of the ocean circulation and land surface physics that are as complicated as the atmospheric GCM itself. Another aspect that differs widely among extant GCMs is the treatment of the middle and upper atmosphere. Most GCMs have been developed specifically to study tropospheric climate problems, and so typically have a domain extending from the ground up to the middle stratosphere, and strongly concentrate the numerical resolution in the troposphere. However, there are now several models that make an attempt at simulation of the mesospheric and even lower thermospheric circulation. The higher a GCM extends, the greater the complications that need to be considered in formulating the model, particularly in the radiative transfer calculations.

The development of coupled dynamics/chemistry GCMs is now underway, as well (e.g. Rasch et al., 1995). The main focus of such models is on the stratosphere where the chemical and dynamical transport timescales for ozone become roughly comparable. In the mesosphere the photochemistry generally acts on much shorter timescales than the transport, so the need for coupled dynamical/chemical models is less compelling. However, there is at least one important chemical transport problem in the mesosphere, i.e. the downward flux of NO in the polar night, which can have a significant effect on stratospheric ozone chemistry (e.g. Barth, 1995).

The whole approach of constructing such complicated simulation models is somewhat unusual in physical science, which typically takes a more directly reductionist tack. It is certainly not a priori obvious that GCMs should be particularly useful tools to gain insight into the detailed dynamics of the atmosphere. The difficulty, of course, is that the results of such models tend to approach the complexity of the actual

atmospheric flow. On the other hand. there is a critical advantage to analyzing a model, in that all the atmospheric variables are known, in contrast to the imper- fectly observed real atmosphere. In addition, the modeller can perform controlled experiments with the model atmosphere that can involve large perturbations which would be impossible for scientists to impose in the real atmosphere (doubtless fortunately for mankind!). In fact, the analysis of GCM simulations has turned out to be an extremely fruitful enter- prise for meteorologists. Three decades have now passed since the publication of the results from the first true GCM (Smagorinsky et al., 1965) and in the ensuing years there have appeared hundreds of papers dealing with GCM results. Of course, progress in dynamic meteorology has also required the development of simpler models as well, and much of the diagnostic work on GCM results has been guided by expectations based on simpler theoretical models.

2. GENERAL CIRCULATION MODELS

Here I will try to describe the issues that have had to be addressed as GCMs have been developed. I will first consider the numerical solutions to the Navier- Stokes equations, and then discuss the problem of incorporating effects that are expected to occur on scales smaller than can be resolved explicitly in a practical numerical model. This section concludes with a list of some of the extant GCMs which have been developed for middle atmospheric studies, and a brief description of two of the more extensively analyzed of the models. There is at present no publication that has a complete coverage of the subject of general circulation modelling, although the book by Wash- ington and Parkinson (1986) perhaps comes closest. Haltiner and Williams (1980) present a more detailed discussion of numerical methods applied to global atmospheric models.

2.1. Governing equations and numerical methods of solution

Global GCMs typically solve what meteorologists refer to as the ‘primitive equations’ which are basically the Navier-Stokes equations for an ideal gas, simplified to incorporate the hydrostatic approximation. The equations are written in the frame of the rotating earth (and thus must include the Coriolis terms in the momentum equations) and the full spherical geometry is considered (deviations of the geoid from sphericity have not been considered thus far). The presence of minor constituents is normally treated by simply including a continuity equation for the mixing ratio

Comprehensive meteorological modelling of the middle atmosphere: a tutorial review 1593

of each species considered. The effect of the changes in trace constituent concentration on the molecular weight of air is normally not considered, except for the case of water vapor. (See any meteorological text for the details of the: standard treatment of the effects of water vapor in the hydrostatic equation, e.g. Holton, 1993.) Since meteorological models have generally considered only the region well below the turbo- pause, the effects of diffusive separation of long-lived constituents have not been treated. Similarly, meteorological models have not considered the effects of terrestrial magnetic and electric fields on the circulation.

The net result is lhat solutions must be found to a set simultaneous equations involving time and space derivatives: two horizontal momentum equations, the hydrostatic relation, the thermodynamic equation, equation of state, a continuity equation for the total mass and continuity equations for the water vapor mixing ratio and the mixing ratio of each of the other trace constituents included.

The equations are generally written with pressure or some scaled version of pressure (such as the ‘sigma’ coordinate discussed below), rather than geometrical height, as the verti’cal coordinate. A version of the primitive equations with pressure as the vertical coordinate is given below (derivations can be found in standard dynamical meteorology textbooks, e.g. Holton, 1993)

d uv tan 8 ,u-fv+-p= a &j&-E (2.1)

u* w $v+fui-atanD= -a-FY (2.2)

aqb RT _=- ap P

(2.3)

Cp& T- ‘y = Q-uF,-vF,+H (2.4)

1 ~ -qvcos0)+ acostl ae A;+$=0 (2.5)

where d/dt is the material derivative, i.e.,

(2.6)

Here p is the pmssure, 0, 1 are the latitude and longitude, U,U, are the zonal and meridional wind components, o is the vertical velocity in pressure coordinates (dp/dt), T IIS the temperature, R is the gas constant for air, C, is the specific heat of air at constant pressure, 4 is the geopotential, f is the Coriolis parameter. The terms F, and Fy are the drag due to

fictional effects (molecular viscosity), and Hrepresents the effects of heat conduction. The term Q is the diabatic heating rate per unit mass, which is normally taken to be the sum of the convergence of the total radiative flux and the rate of latent heat release in condensation of water vapor.

Note that the thermodynamic equation has been written including the term - (uFx+uF,,) which represents the heating that results from the dissipation of kinetic energy by friction. Most GCMs ignore (or grossly approximate) this contribution to the thermodynamic equation. This is reasonable in the lower atmosphere, but there is some interest in including this term in models of the upper mesosphere and lower thermosphere, a point that I will return to later. The equations given above need to be supplemented by a continuity equation for water vapor and continuity equations for any trace constituents included.

The basic problem of finding approximate solutions to these governing equations is essentially the same as in time-dependent simulation models of the upper atmosphere (e.g. Roble et al., 1988). Some common features that appear in the approaches have been employed to numerically approximate these partial differential equations in all (or nearly all) extant meteorological GCMs. In particular, the time derivatives are generally replaced by a simple centered finite difference, so that the fields at predicted time to+ At depend on the fields at the two previous discrete time levels t,, and t,-- At. The coupling between the time levels can be formulated to allow explicit solution (e.g. second-order ‘leapfrog’) or can involve implicit treatment of at least some terms (e.g. Haltiner and Williams, 1980). The vertical derivatives have also generally been replaced by finite differences. The approximation of horizontal derivatives in the governing equations provides the most variation among different models. The so-called ‘gridpoint’ models employ finite difference approximations. This is a fairly straightforward procedure in principle, but there are some issues that must be addressed in practice. In particular, the most appropriate arrangement of gridpoints on a sphere is not obvious. The largest time-step that allows stable numerical integration is generally proportional to the shortest horizontal scale resolved, so a simple latitude-longitude grid is extremely inefficient. The most common solution to this problem is to retain the latitude-longitude grid, but to perform some low-pass zonal filtering of fields at high latitudes to leave the effective zonal resolution fairly constant with latitude.

The other popular approach to approximating horizontal derivatives is the spectral method. In this method, the variables on any model horizontal surface

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are written as sums of a finite set of spherical harmonics “I”, (where m is the zonal wavenumber and Z-m is the number of zero crossings from pole-to- pole), and the governing equations are recast as equations for the coefficients of the various harmonics. In practice, this method becomes feasible for nonlinear equations by use of the ‘transform method’, in which the nonlinear products of variables are computed by transforming to a grid in physical space, performing the necessary multiplication of the values of the fields on this grid, and then transforming back to get the spectral coefficients of the nonlinear term. The aspects of the model integration that depend only on the vertical coordinate (radiative transfer, subgrid-scale vertical mixing, exchanges between the atmosphere and the surface) are normally performed on the physical space grid as well.

An important recent development is the so-called ‘semi-Lagrangian’ approximation for advective terms (e.g. Ritchie, 1991). This can be implemented directly in gridpoint models or by using the transform grid of a spectral model. The basic idea here is to imagine the gridpoints imbedded in a continuous flow. The change in some quantity (heat, momentum, tracer concentration) at a particular gridpoint x, due to advection between t, and t,+At is the difference of the value at the gridpoint at t, and the value at the parcel that will arrive at the gridpoint at t,+At. The position of this parcel at time t can be approximated by x’ = x0-v(x,, t,)At, where v is the vector wind. The value of the quantity of interest at the position x’ is estimated by interpolation from the grid values at t,. This has some advantages over purely Eulerian advection schemes in terms of formal accuracy, and it can be integrated stably with an arbitrarily large time- step. In practice there are details that vary greatly from model to model. The semi-Lagrangian scheme can be used in three dimensions, or employed just to compute the horizontal contribution to the advection. The scheme can be made implicit by making x’ depend on the velocity at t,+At. In GCMs the semi- Lagrangian treatment could be used for all variables including heat and momentum, but is often restricted to computing the advection of water vapor and other trace constituents.

The treatment of the entire global atmosphere elim- inates the need to consider lateral boundary conditions in a GCM, but solution of the primitive equations requires the imposition of appropriate boundary conditions at the top and bottom of the atmospheric domain considered. At the top the usual approach is to impose a condition of no vertical motion (ie. w = 0) at the highest level. This is referred to as the ‘rigid lid approximation’. Often the condition

is applied at a level formally at zero pressure, however, it has been shown that this is still equivalent to imposing a lid on the atmosphere at finite pressure (Lindzen et al., 1968). This has the effect of creating spurious reflections of upward propagating waves incident on the upper boundary in the model. Most climate GCMs have ignored this problem, but some models with significant resolution in the middle atmosphere have been formulated to reduce downward wave reflections by including an imposed linear damping of the flow at the top level or levels.

One complication in a realistic meteorological GCM that is avoided in simplified models treating only the middle atmosphere and/or the thermosphere, is the necessity to incorporate the effects of surface topography. The usual way of doing this is to recast the equations in terms of a vertical coordinate that is terrain-following. The simplest instance of this is the so-called ‘sigma’ coordinate o = p/p*, where p* is the surface pressure at a particular location. The obvious advantage of this is that the surface is always e = 1, and so the boundary condition of no mass flux into the surface is simply that the vertical sigma velocity (do/&) be zero at the model level at e = 1. For models treating both the troposphere and the middle atmosphere there are important advantages in formulating the governing equations in terms of a coordinate that is terrain-following near the ground and purely isobaric at high levels. This could be accomplished in a number of ways, but the most common is the so-called sandwich coordinate defined as (e.g. Fels et al., 1980)

c = (P-PJI(P* -P,) for P > pi (2.7)

0 = (P-PMP,~-P,) for P < pi (2.8)

wherep* is that actual pressure at the surface (a function of geographic location and time) and ps is a constant (taken to be a typical value for p*). In this case the coordinates are isobaric above the interface pressure pi The vertical velocity in this system is the material derivative of r~, which above the interface level will be just proportional to the pressure velocity cu. The use of isobaric coordinates at high levels appears to produce a better behaved numerical simulation and facilitates analysis of model results above the interface level.

2.2. Subgrid-scale physical processes

All numerical models of fluid systems any more complicated than a simple laminar flow must deal with a nontrivial subgrid-scale closure problem, i.e. the necessity to incorporate the effects of processes that occur on time and/or space scales too small to resolve explicitly. For many problems in turbulent flow the


assumption is made that the subgrid-scale motions simply act as an effective viscosity on the resolved flow. While GCMs typically include some kind of subgrid-scale ‘turbulent viscosity’, the complexity of atmospheric circulation requires the inclusion of a much more involved set of parameterizations. Such parameterizations can be divided into those that involve transfers between the atmosphere and the earth’s surface and those that are purely internal to the atmosphere. In formulating the internal parameterizations it is necessary to consider the strong spatial anisotropy apparent in the atmospheric flow. In particular, subgrid-scale mixing in the vertical is generally treated very differently from that in the horizontal direction. A key point that must be considered is that vertical mixing should be inhibited by the large-scale static stability, and that conditions where gravitational instability might be expected to develop will presumably lead to rapid mixing in the vertical. In the absence of latent heat release due to water vapor condensation, this can ‘be handled by straightforward, if somewhat ad hoc, methods. One approach is simply to include a vertical diffusion (of heat, momentum, moisture and any other trace constituents) that is a function of the local Richardson number of the resolved flow. Anotlher possibility is the so-called dry convective adjustment (DCA), which is a procedure applied at the end of each time-step of the integration. In applying the DCA at a particular gridpoint, the temperatures in the column above are checked to see that the computed lapse rate is gravitationally stable. At any point where there is an unstable layer, the adjustment is applie’d, i.e. heat is assumed to be mixed among the levels in the unstable layer until gravitational stability is restored. In addition, many models include some form of enhanced vertical mixing near the ground.

The problem of parameterizing convection in the presence of moisture condensation is one of the most challenging faced by meteorological modellers. Obser- vations of actual convective clouds show a complicated pattern of intense updrafts with entrainment of moist air at low l’evels, detrainment of dry air near the cloud tops and broad regions of (cloud free) down- drafts. Conceptually, perhaps the simplest parameterization of moist convection is the so-called moist convective adjustment (MCA), which is a generalization of the DCA discussed above. Just as for DCA, the MCA is applied at the end of each time- step of the integration. The MCA procedure begins with the identification of those levels at which the predicted water vapor concentration exceeds the saturation value (a function of temperature and pressure). The excess water vapor is assumed to immediately

condense out and the associated latent heat release warms the air. After this step, the lapse rate at saturated levels is checked to see if it exceeds the gravitational stability threshold expected for a saturated atmosphere. If so, a mixing of heat and moisture between adjacent levels is applied to just stabilize the lapse rate. This procedure can leave some levels super- saturated, and so the MCA is normally iterated until all supersaturation is eliminated and all saturated layers are gravitationally stable.

There are many other parameterizations of moist processes that have been employed in GCMs. Some are close in spirit to the MCA scheme (e.g. Betts, 1986) in focusing on the role of convection in restoring a gravitationally-stable state, while others emphasize the dependence of the convective heating within an atmospheric column on the near-surface moisture convergence (Kuo, 1974). Recent work on the parameterization of moist convection in numerical models is reviewed in Emanuel and Raymond (1993).

One important process for the atmospheric circulation that needs to be parameterized is the drag on the atmosphere exerted by the surface. Note that the only part of the momentum exchange between the earth and the atmosphere that can be explicitly resolved in a GCM is the so-called ‘mountain torque’, i.e. the drag on the atmosphere due to the upstream- downstream pressure differences across the topographic features. There is also significant drag from frictional effects (note that it would take very high vertical and horizontal resolution to explicitly model the molecular viscous boundary layer over the earth’s surface) as well as from mountain torque associated with topographic features of unresolvably small horizontal scale. Typically it is assumed that all the unresolved drag can be approximated by the following formula:

r = -C,plvlv (2.9)

where r is the horizontal stress on the atmosphere across its lower boundary, p is the air density at the surface, v is the horizontal wind velocity at the lowest model level and Co is a dimensionless ‘drag coefficient’. CD in general should depend on the rough- ness of the surface (e.g. one expects more drag for flow over a forest than over grasslands) and can also depend on the resolved static stability and shear in the atmosphere near the ground. In the usual approach this stress is included in the horizontal momentum equation of the lowest model level (effectively assuming that the stress is distributed throughout the mass of the lowest resolved layer).

The exchanges of heat and moisture between the

1596 K. Hamilton

surface and the atmosphere generally are also parameterized with analogous formulae. Thus the sen- sible heat flux into the lowest layer will be assumed proportional to the horizontal wind speed at the lowest model level times the difference in temperature between the ground and the lowest level. Over the ocean the evaporation of water vapor into the lowest level is taken to be proportional to the wind speed and to one minus the relative humidity of the air in the lowest model level (i.e. the drier the air, the stronger the evaporation from the surface). Over land such a formula must be modified to account for the degree of soil wetness. This requires either a specification of the soil wetness or a prognostic model to compute the soil wetness at each gridpoint (including effects of precipitation, evaporation and run-off from adjacent gridpoints). I want to emphasize that the geographical modulation of the soil wetness is actually quite important for determining the precipitation in GCMs (e.g. Koster and Suarez, 1995). The tropospheric heating associated with precipitation is thought to be a very significant excitation for the vertically-propagating waves that dominate the circulation of the mesosphere and lower thermosphere (e.g. Manzini and Hamilton, 1993). This is an instance of possible significant upper atmospheric influence of what one would be tempted to regard as a minor detail of the tropospheric formulation of a model, and is an indication of the challenge inherent in formulating comprehensive models of the global atmosphere.

One aspect of the subgrid-scale parameterization problem that has come to be appreciated in recent years is related to the role of internal gravity waves in exchanging mean flow momentum between different altitude regions in the atmosphere and between the atmosphere and the surface. An important example of this is the effect of topographically-forced gravity waves. In stably-stratified conditions, part of the force exerted on the atmosphere by the topography will act to generate vertically-propagating internal gravity waves. Such waves will contribute to the mean flow momentum at the levels at which they are dissipated (see Section 4.2, below). Thus the topographic influence on the atmospheric momentum budget is expected to have a nonlocal component. Some of this effect will be explicitly modelled in a GCM, of course. However, the effects of topography in generating subgrid-scale waves needs to be parameterized. Definitive calculations or observations concerning this effect are not available, but it is certainly possible that the subgrid-scale topographic wave effects are important for GCMs, and many current climate models incorporate some kind of topographic gravity wave parameterization (e.g. Palmer et al., 1986; McFarlane,

1987). Of course, topography is only one possible excitation for gravity waves in the atmosphere, and subgrid-scale waves forced by other, nonstationary sources can also have an effect on the mean flow. Such nonstationary waves are thought to be particularly significant in the general circulation of the middle and upper atmosphere (e.g. Lindzen, 1981; see also Section 4.2, below). The parameterization of gravity wave effects will be discussed in Sections 5 and 6 below.

2.3. Radiative transfer

From the perspective of a meteorologist, the prin- cipal radiatively-active constituents which contribute to the atmospheric heating are water vapor, cloud droplets, CO, and O,, and almost any atmospheric GCM will have radiation codes that treat both the longwave and shortwave effects of all these constituents. Of course, in models of the mesosphere and lower thermosphere, the absorption of solar radiation by 0, must also be included. The effects of other aerosols (notably stratospheric sulfuric acid droplets resulting from volcanic eruptions) and other trace gasses such as N20 and CH, are ignored in most models, but could be significant in determining the precise temperature in the stratosphere (e.g. Wang et al., 1991). The shortwave (solar) and longwave (terrestrial) parts of the radiation problem are treated separately. The shortwave heating depends on the solar zenith angle, the distribution of absorbing constituents and reflec- tivity of both the surface and each of the atmospheric levels, and is nearly independent of the atmospheric temperature. By contrast, the longwave heating depends strongly on the atmospheric temperature structure as well as the distribution of absorbers. Typi- cally the solar absorption is parameterized as a function of absorber path length using formulae that accurately fit detailed line-by-line calculations (e.g. Lacis and Hansen, 1974). The longwave heating rates will normally be computed within a band model approximation for the transmissivity between atmospheric layers, and with the assumption of local thermodynamic equilibrium to compute the source function. Liou (1980) discusses the basic approaches for the radiative transfer algorithms typically used in meteorological models. The usual assumptions made in standard algorithms break down significantly above about 70 km. In particular, the assumption that the solar energy absorbed by an atmospheric constituent appears immediately as heating (i.e. an increase in molecular translational kinetic energy) is not valid at high altitudes, and the effects of chemical heating, airglow and chemiluminescence of the excited products of solar absorption need to be considered (e.g.


Mlynczak and Solomon, 1991). In addition, the assumption of local thermodynamic equilibrium in the computation of longwave emission becomes invalid at high altitudes. These effects make the straightforward application of current meteorological GCMs near and above the mesopause rather problematic.

A very significant complication in the radiative transfer in a full GCM is the treatment of cloudiness. It is known that clouds play an extremely important role in both the longwave and shortwave radiation budgets of the troposphere. Unfortunately, both the calculation of radiative fluxes within clouds and the actual determination of an appropriate three-dimensional cloudiness field are extremely difficult problems. Typically GCMs will use some kind of simplified cloud geometry, such as allowing only three layers of cloudiness (i.e. for each grid box a specification of how much of the area is covered by ‘low’, ‘middle’ and ‘high’ clouds) and making some assumption about how much spatial overlap occurs between the cloud layers within the grid box. The cloud fractions can be specified on the basis of climatological observations or can be predicted within the model with some parameterizati on (typically depending on the grid-scale relative humidity, the grid-scale vertical velocity and the static stability). At this point no modeller would claim a credible first principles approach to the cloudiness problem, and any scheme must be tuned considerably to assure a reasonable climate simulation. This remains the most important single challenge in climate modelling.

2.4. Examples of current GCMs with significant middle atmospheric coverage

Studies of the stratospheric circulation in troposphere/lower stratosphere GCMs began with the pion- eering papers of Manabe and Hunt (1968), Kasahara and Sasamori (1974) and Manabe and Mahlman (1976). There are now comprehensive GCMs being developed primarily for middle atmospheric studies at research centers in several different countries. Table 1 is a (no doubt incomplete) list of such models. Typi- cally these GCMs a:re being derived from earlier tropospheric/lower stra.tospheric climate models. Here I want to give a brief description of the GFDL and NCAR models listed in this table. In this paper I will be largely reviewing results from the ‘SKYHI’ GCM developed at GFDL. This is the middle atmosphere model with the longest history of development and is perhaps the one that has been most extensively applied in middle atmospheric studies. The NCAR Com- munity Climate Model (CCM) is also briefly described, since it presents some interesting contrasts

with SKYHI in terms of model formulation and results, and because long integrations with various versions of this model have also been extensively studied.

The GFDL SKYHI GCM is designed to simulate the global atmosphere from the ground to the mesopause (Fels et al., 1980; Mahlman and Umscheid, 1987; Hamilton et al., 1995). In SKYHI the primitive equations are discretized on a horizontal grid. I will mention three different versions with 3 x 3.6”, 2 x 2.4” and 1 x 1.2” latitude-longitude resolution. A sandwich vertical coordinate with interface pressure of 353 mb is employed. In all published versions thus far 40 levels in the vertical have been used, with the altitude spacing between levels gradually increasing with height. The mesospheric levels are at 0.0096, 0.0308, 0.0697, 0.131, 0.222 and 0.349 mb, corresponding roughly to heights of 80,73,68,63,59 and 55 km. At the top of the domain the ‘rigid lid’ boundary condition is used along with a linear damping applied to eddy motions at the highest full model level (see Fels et al., 1980). The model is integrated with explicit leapfrog time differencing using time steps of 75, 180 and 225 s for the 3 x 3.6”, 2 x 2.4” and 1 x 1.2” versions, respectively. A realistic topography and land-sea distribution are employed. Figure 1 shows the topography over part of North America as rep- resented at 3 x 3.6” and 1 x 1.2” resolutions. The model includes a sophisticated computation of the longwave and shortwave heating rates. Mean ozone amounts are prescribed, but locally the mixing ratio above 35 km is allowed to vary linearly with temperature in order to account for the photochemical acceleration of radiative eddy damping (see Fels et al., 1980). The cloud field used in the radiation code is also prescribed. The radiation code includes absorption of solar radiation by clouds, H20, 03, O2 and CO* and cooling through IR emission by clouds, H20, CO, and OX. The treatment of the radiative transfer is based on local thermodynamic equilibrium and so ignores non- LTE effects that may be very significant above -70 km. In the standard version of SKYHI the solar heating rates are computed as diurnal mean values. There is also a version with a diurnal cycle (with heating rates computed each hour; see Hamilton, 1995a).

The SKYHI model includes a representation of the hydrological cycle, with standard parameterizations of evaporation and moist convective adjustment (see Manabe, 1969). A dry convective adjustment and a Richardson number-dependent vertical diffusion (Levy et al., 1982) are also employed. The horizontal subgrid-scale mixing of heat, momentum, moisture and other tracers is parameterized as a second-order diffusion with coefficient proportional to the local horizontal flow deformation (Andrews et al., 1983).

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Table 1. Some current comprehensive troposphere-stratosphere or tropospherestratospheremesosphere GCMs, with selected references

Center References Comments

GFDL, Princeton, U.S.A.

NCAR, Boulder, U.S.A.

NASA/GISS, New York, U.S.A. MAM Project, Canada

Free U. Berlin, Germany

MPI Hamburg, Germany

Meteo-France, France

BNMRC, Australia

MRI, Japan U.K. Universities, United Kingdom

Fels et al. (1980); Hamilton et al. (1995)

Boville (1986, 1991, 1995)

Rind et al. (1988a,b) Shepherd (1995)

Pawson et al. (1991)

Manzini (1994)

Cariolle et al. (1993)

Hunt (1991)

Shibata and Chiba (1990) Jackson and Gray (1994)

Grid point model with 40 levels in the vertical from the ground to the mesopause run in some cases with very high horizontal resolution Spectral model with domain extending above the mesopause Grid point model extending to 1 mb Spectral model with domain extending above the mesopause Spectral model with domain extending into the mesosphere Spectral model with domain extending into the mesosphere Spectral model with 30 levels from the ground to 80 km Mixed spectral-grid formulation with 52 levels from the ground to 100 km, but with quite coarse horizontal resolution Spectral model with 23 levels up to 0.05 mb Spectral model extending to near mesopause

In the standard version no attempt is made to parameterize the effects of subgrid-scale gravity waves. In the experiments described here a realistic climatological annual cycle of the sea surface temperature is pres&bed. Surface temperatures and soil wetness over land are calculated for each time step using a simple prognostic soil model.

The NCAR CCM is similar to SKYHI in its overall ambition to model the full troposphere-stratosphere- mesosphere system (Boville, 1986, 1991, 199.5), but is rather different in its numerical formulation and physical parameterizations. The horizontal dis- cretization in CCM is done using the spectral method and the time integration is done with an implicit scheme. The middle atmosphere version of the CCM has been run at spectral resolutions varying from T21 to T106, where this notation refers to the largest total horizontal wavenumber, 1, included in the truncated spectral representation. The vertical resolution has also been varied for different applications; in the current standard CCM version described in Boville (1995) there are 44 levels up to about 0.01 mb. The current version of the CCM uses a semi-Lagrangian approach to the calculation of advective terms for water vapor and any chemical tracers (but not for heat or momentum). The horizontal mixing is treated by a linear fourth-order diffusion and (like SKYHI) a Rich- ardson number-dependent vertical diffusion is included. The CCM has a different moist convection scheme than SKYHI, and also prognostically determines the cloud amounts used in the radiative transfer

calculations. The CCM differs from SKYHI in including a special treatment of the flow in the planetary boundary layer (roughly the lowest km or so of the atmosphere) rather than simply applying the same vertical mixing parameterization at all levels (as in SKYHI). Unlike SKYHI, the CCM includes a simple parameterization of the subgrid-scale topographic gravity waves (following McFarlane, 1987).

3. TROPOSPHERIC SIMULATION

Much effort has been expended on comparing the tropospheric climate simulated in GCMs with observations, as well as intercomparing results for various models. The standard test is to run a model with prescribed realistic sea surface temperatures over a period of at least several years and then compare long- term means (and possibly interannual variances) of various quantities. One very extensive study of this sort is described in Boer et al. (1992); this involved comparisons of a large number of models from research centers in several different countries. The basic quantities such as the zonal mean temperature and zonal wind fields tend to be reasonably well simulated in most models. Figure 2 shows the upper troposphere (200 mb level) zonal wind averaged over two December-February (DJF) periods in an integration with the 2x2.4” SKYHI model, compared with a long-term mean observed climatology. The model result in this case displays the basic meridional structure of dominant tropical easterlies and strong extra-

1599

Fig. 1. The topography used in the SKYHI model shown over an area roughly coincident with the continental U.S. for the 3 x 3.6” (top) and 1 x 1.2” (bottom) resolutions. The highest point shown in the

1 x 1.2” topography shown is 3000 m, while for the 3 x 3.6” topography the highest point is 2418 m.

tropical westerlies in both hemispheres (but particularly strong in the winter extratropics). The concentration of the most intense westerlies in geo- graphically localized jets is also well captured in the model simulation.

A particular challenge for models is the simulation of the precipitation field, since this very directly involves the model subgrid-scale parameterizations and their interaction with the large-scale dynamics (which control the moisture convergence). Figure 3 shows annual mean results for total precipitation from 2 yr of a 1 x 1.2” SKYHI simulation, compared with observations (results shown here only over land where fairly reliable observations are available). Many of the geographical variations in the observed precipitation are captured in the model simulation. A nice indication of the variability in tropospheric climate simulations among curre:nt models is given in the paper of

Boer et al. (1992) who present zonally-averaged June- August (JJA) precipitation obtained from a large number of individual GCMs from various research centers throughout the world. The general pattern of the simuIated field in each of the models is realistic, at least to the extent that each displays a near-equatorial maximum and subtropical minima in each hemisphere. However, the actual values differ quite substantially among the models (by as much as a factor of 2 at some latitudes).

An important aspect of the precipitation (and associated latent heating) in GCMs is the rather com- plex temporal variability that can be simulated. The space-time variability of tropospheric latent heating does greatly affect the spectrum of the vertically-propagating wave field in the middle atmosphere (e.g. Manzini and Hamilton, 1993). Figure 4 shows a time- longitude section of the equatorial precipitation dur-

1600 K. Hamilton

Fig. 2. December-February mean 200 mb zonal wind climatologies from (top) observations (Schubert et al., 1990) and (bottom) from the 2 x 2.4” SKYHI model. The contour interval is 10 m SF’ and dashed contours denote westward winds. The results from the SKYHI model represent averages over two con-

secutive years of integration.

ing 4 days of integration with the 2x2.4” SKYHI model. Evident in this figure is a qualitatively realistic organization of the precipitation into weather systems. (Note that this integration did not include a diurnal cycle of solar radiation). A comprehensive quantitative comparison of the statistical space-time precipitation variability in GCMs with observations apparently has not been attempted thus far.

4. BASIC ISSUES IN STRATOSPHERIC DYNAMICS AND

TRANSPORT

4.1. The zonal-mean circulation in the extratropics The simulation of the most basic aspects of the

global-scale tropospheric circulation have been handled reasonably well by most GCMs. Obtaining a reasonable simulation of the wind and temperature climatology for the stratosphere is inherently more

difficult than the corresponding problem for the troposphere. In the troposphere there is a strong vertical coupling of the temperature to that of the surface through convective processes, and s-given realistic prescribed ocean surface temperatures-a reasonable model should be able to simulate a fairly realistic time- mean tropospheric circulation.

In the stratosphere the direct coupling to the surface is not significant, and the temperature structure can be thought of as resulting from a competition between the in situ radiative heating which tries to create a state with very cold winter polar temperatures, and dynamical processes that drive the temperatures away from this state. Fels (1987) reports on an experiment in which the radiative code from SKYHI was marched through an annual cycle with the tropospheric temperatures effectively prescribed to fairly realistic values (see Fels, 1987, for details). This kind of cal-


OBS

............

............ .... ....

Cl

lox 1.2’ SKYHI

>6 Fig. 3. Annual-mean precipitation rate climatologies from observations (top) and from the 1 x 1.2” SKYHI model (bottom). The results for the model are based on 2 yr of integration. The key to the shading is labelled in mm day-‘. Results shown only for land areas equatorward of 60” latitude. The observations

are from Legates and Willmott (1990).

1602 K. Hamilton

6d0E 12;OE woo

a- 16 16-32 > 32

Fig. 4. Latitude-time section of precipitation rate along the equator from four days of integration with the 2x2.4” SKYHI model. Plotted are slightly-smoothed 6-h averages at each gridpoint. The key to the

shading is labelled in mm day-‘.


Fig. 5. Latitude-height section of the temperature (“K) in January simulated in the purely radiative model of Fels (1987).

culation represents a generalization of the usual notion of a radiative equilibrium state (normally defined as the temperature structure that results in no net radiative heating). The result of Fels’ calculation in January is shown in Fig. 5. This radiative state is extremely unrealistic, with winter polar temperatures much colder than observed (by N 100°C near 1 mb). The zonal-mean zonal winds that are in balance with the radiative temperature state are shown as Fig. 6.

15’N 25’ON 3s;N 46-N 55-ON f&-N 76-N &N Fig. 6. Zonal wind fielsd that is in gradient-wind balance with the temperature field shown in Fig. 5. The contours are lahelled in m s-’ and positive values represent eastward

winds. Results shown only poleward of 15”N.

Again the result is extremely unrealistic with eastward winds of over 300 m s-’ appearing near the mesopause (see Fig. 9 below for a view of the observed NH winter climatology).

The net effect of dynamics on the winter extratropical middle atmospheric circulation is to act as a drag on the strong eastward jet that would be forced by radiation alone. The effects can be diagnosed with the following approximate equations for the zonally averaged circulation:

foEa -i-i @(UW ,+(--& ;(;;;;cosro)) (4.1)

fii+ctane= -L$$ a (4.2)

-+- P

& Tjj(mose) 7%

-cc(T- Trad)- & $(mcose) (4.3)

~-$ocose)+~m=O (4.4)

where S is a measure of the vertical stability (normally S>O),

s=E-_!!I C,P ap (4.5)

1604 K. Hamilton

and where the overbar denotes the zonal mean and primes denote deviations from the zonal mean. In these equations the time-dependence of both the wind and temperature have been neglected and the only nonlinear terms are considered the divergence of the eddy momentum and heat fluxes. In addition, the diabatic heating has been crudely approximated by a linear expansion of the deviation of temperature from the purely radiative equilibrium state (Trad). In the absence of friction and eddy fluxes, the only solution is pure radiative equilibrium (F = Trad) along with a zonal wind field in geostrophic balance with Trad. The effects of the eddy fluxes can be regarded as effectively forcing the mean meridional flow and thus driving the circulation away from the radiative equilibrium solution. In particular it seems that in the winter extratropical stratosphere there must be a strong sinking motion (i.e. w > 0) to account for the fact that temperatures are much warmer than in the purely radiative state.

The equations (4.14.4) are obviously not sufficient to completely model the circulation, but they are thought to express the dominant balances in the extratropics of the middle atmosphere. It is convenient to combine the mean-flow effects of the eddy momentum and heat fluxes through the use of the ‘transformed- Eulerian’ formalism introduced by Andrews and McIntyre (1976). In particular, if one defines the Eliassen-Palm (EP) flux with meridional (F@) and vertical (Fp) components, as

Fe = -a COS~U'V' (4.6)

FP = -acose (

m-f:m >

(4.7)

and a transformed meridional circulation as

a c- v* = o_- ‘TV

i > ap s (4.8)

C Q)“=Q- A ( >( s &$mcose)) (4.9)

then the zonal momentum equation can be rewritten as

(4.10)

and the thermodynamic equation becomes

e W* = - (C,S) (4.11)

Note that the sign convention for EP flux follows

that of Dunkerton et a/. (1981) which has become fairly standard (e.g. Andrews et al., 1987), but differs from that in the original paper by Andrews and McIn- tyre (1976). The transformed meridional circulation can also be shown to satisfy the standard continuity equation (i.e. replace v and o in equation 4.4 with v* and w*).

The system of equations 4.64.11 represents a considerable conceptual simplification-the transformed meridional circulation is determined entirely by the EP fluxes (through equation 4.10 and the continuity equation) and the transformed meridional circulation determines the zonal mean heating (and hence how far the temperature is driven from the radiative equilibrium) through equation 4.11. The right hand side of equation 4.10 is referred to as the ‘EP flux divergence’ (EPFD) and is the most relevant measure of the eddy- induced driving of the zonal mean circulation. Of course, the derivation presented here has started with the approximate equations 4.14.4. The simplest generalization of this scheme is to include the tendency term for the zonal-mean zonal wind in equation 4.1. If the effects of the radiatively-driven residual circulation were to be negligible (say because the radiative timescales were very long or f were very small), then the EPFD would directly force a mean-flow acceleration. In practice, any EPFD associated with the eddies ends up producing a response involving both mean-flow acceleration and an adjusted residual meridional circulation.

The ideas of EP flux and residual meridional circulation have been extended to the full primitive equations (e.g. Andrews and McIntyre, 1976; Andrews et al., 1983, 1987) and the complete expressions for EP flux and EPFD are used in the analysis of model results presented in Section 5, below. Also discussed by Andrews and McIntyre (1976) and Andrews et al. (1983, 1987) is another important aspect of the EP flux diagnostic. In particular, it is known that for the case of linear, steady-state waves the EPFD is zero outside of regions of forcing and dissipation, and thus the EP flux itself is conserved under these cir- cumstances The EP flux can thus be used to diagnose the wave dissipation (including the effective dissipation arising from nonlinear behavior). The notion of EP flux as a measure of the propagation of the activity of wave disturbances can be made more precise, at least under conditions where the mean flow varies slowly in space and time (e.g. Andrews et al., 1983, 1987).

4.2. Effects of vertically-propagating gravity waves

One important contribution to the EPFD in the middle atmosphere comes from vertically-pro-


pagating inertia-gravity waves. It has been established for some time that much of the variability on timescales from minutes to days observed in the middle and upper atmospheric wind and temperature fields can be attributed to inertia-gravity waves propagating upward from the tro:posphere (e.g. Hines, 1960; Fritts, 1984). The possible effects of such waves on the general circulation have also been considered in models of various levels of sophistication (e.g. Hodges, 1967; Lindzen, 1981). The basic issues can be understood in the context of a very simple model for waves (e.g. Plumb, 1977; Lindzen, 1981). In particular, it can be shown that for a linear, 2-D (zonal and vertical directions), steady-state, monochromatic plane wave in a slowly-varying background state, subject to a linear dissipation with rate CI, the upward flux of eddy zonal momentum is given by

7 U’W (z) = ~‘0 ‘(z,) exp (I( - a/ W,(z))dz) (4.12) 2”

where z is the altitude and W, is the vertical group velocity of the wave, which in general will be a function of altitude. Note that this is consistent with the notion that the EPFID should be zero for. steady linear waves in the absence of dissipation (i.e. tl = 0 in this simple example). For the case with no rotation

B’, = (i~c)~/Nk (4.13)

where k is the zonal wavenumber, c is the zonal phase speed relative to the ground, P is the mean zonal wind and N is the usual Brunt-Vaisala frequency. A key point to notice in equations 4.12 and 4.13 is the dependence on the intrinsic phase speed (n-c). A straightforward analysis of the polarization relations for

I gravity waves also shows that the sign of u o depends on the sign of (n - c) Thus upward-propagating waves with eastward (westward) intrinsic phase speed produce an eastward (westward) contribution to the EPFD. Lindzen (1’981) pointed out the important consequences of these facts for the general circulation of the middle atmosphere. For example, in the winter extratropics the radiative processes try to establish strong mean eastward winds (e.g. Fig. 6). If one imagines that there is a spectrum of waves emerging from the troposphere, roughly balanced between eastward- and westward-propagating components, then the eastward components may be expected to be pref- erentially absorbed at lower levels, while the westward components (with larger intrinsic phase speed) will dominate at higher levels. The net result should be a strong westward eddy forcing of the mean flow in the winter mesosphere, that may help explain the fact that the observed eastward jet closes off in the mesosphere

(rather than increasing in strength with height as a purely radiative model would necessarily predict). The same reasoning can be invoked to explain the closing off of the westward mesospheric jet in the summer extratropics.

The basic notion of a gravity wave contribution to the forcing of the mean flow in the mesosphere is a plausible one, but a detailed global observation of the eddy momentum flux associated with gravity waves is obviously not possible. There have been some attempts to use radar data from individual stations to estimate the U’W’ attributable to high frequency oscillations (e.g. Vincent and Fritts, 1987). In addition, some indirect estimates of the EPFD associated with gravity waves have been attempted (e.g. Hamilton, 1983; Smith and Lyjak, 1985). These results are consistent with the basic notion that there is a very significant EPFD associated with gravity waves in the mesosphere. I will show below in Sections 5 and 6 that comprehensive GCMs can provide a great deal of insight into the role of gravity waves in the general circulation of the middle atmosphere and lower thermosphere.

4.3. Breaking Rossby waves and the ‘surf zone’

Another important source of EPFD in the middle atmosphere is associated with vertically-propagating Rossby waves. Rossby waves are transversely polar- ized waves with a restoring force linked to the horizontal gradients in mean state vorticity (or to be more precise the gradient in the potential vorticity along isentropic surfaces). In a continuously stratified atmosphere Rossby waves can propagate vertically, and in fact the large-scale deviations of the circulation from zonal symmetry seen in the extratropical middle atmosphere can be regarded as an indication of the presence of Rossby waves excited in the troposphere. Many aspects of the large scale circulation in the middle atmosphere can be understood in the context of the theory of small amplitude Rossby waves super- imposed on a zonally-symmetric mean state (e.g. Charney and Drazin, 1961). Particularly significant are the stationary waves forced by flow over topography and other zonal contrasts on the lower boundary. Chamey and Drazin (1961) used linear theory to show that only very large zonal scale stationary Rossby waves (effectively zonal waves 1 and 2) can propagate upward into the strong mean eastward flow present in the winter stratosphere, while stationary Rossby waves are essentially excluded from propagating in the mean westward winds of the summer stratosphere. This accounts for the fact that the monthly-mean flow in the summer middle atmosphere

1606 K. Hamilton

Fig. 7. Schematic view of the potential vorticity as a function of latitude on a potential temperature surface in the winter middle atmosphere. The dashed curve shows the distribution expected from a purely radiative equilibrium solution. The solid curve shows the result of strong mixing of the potential

vorticity in the midlatitudes.

is observed to be nearly a zonally-symmetric westward vortex, while the monthly-mean observed flow in the winter stratosphere is a strong eastward vortex deformed significantly by both wavenumber 1 and 2 perturbations (e.g. Hare, 1968). The deviations from zonal symmetry are also larger in the NH winter stratospheric vortex than the SH winter, due to the much stronger topographic forcing in the NH. The theory of linear Rossby waves can even be used to help understand some of the transient aspects of large- scale waves in the winter stratosphere (Matsuno, 1970, 1971).

In recent years there has been an increasing appreci- ation for the degree of nonlinearity in the Rossby wave field in the middle atmosphere, particularly in the NH winter. McIntyre and Palmer (1984) examined the potential vorticity (PV) computed from analyzed fields based on satellite radiometer temperature measurements. They noted that the pole-to-equator gradient in PV tended to be concentrated in a narrow region around the edge of the polar vortex, which separated the high PV values inside the vortex from lower, more uniformly-mixed values in the mid- latitude region. They supposed that the homo- genization of the PV was accomplished by irreversible mixing associated with strongly nonlinear Rossby waves. McIntyre and Palmer coined the term ‘surf zone’ for the relatively uniformly-mixed region, making an analogy with the mixed region in the ocean near a beach which is produced by breaking surface gravity waves. The basic notion is illustrated sche- matically in Fig. 7, which shows the zonal-mean PV

corresponding to the radiative equilibrium state and then the PV distribution that is produced by vigorous mixing associated with breaking Rossby waves in midlatitudes. The net effect is to produce strong gradients at the edge of the polar vortex and at the equatorward edge of the surf zone. The same general picture can be expected for relatively long-lived trace constituents that have chemical sources and sinks that produce large-scale equator-to-pole gradients. Thus, for example, Leovy et al. (1985) showed a similar behavior in the stratospheric ozone concentrations measured from satellite. Figure 8, adapted from Ruth et al. (1994), shows NH satellite observations of N20 mixing ratio plotted on the 1150°K isentropic surface (roughly 40 km height) on a day in December. The N,O chemistry and large-scale circulation tend to produce smaller values of the N,O mixing ratio near the pole than at the equator. Figure 8 shows that the low values are concentrated in the central vortex and that there is a region along the edge of the vortex with strong gradients. In fact, McIntyre and Palmer speculated that the rather coarse effective horizontal resolution of satellite-derived data such as those shown in Fig. 8 prob- ably underestimates the sharpness of the division of the winter stratospheric circulation into a polar vortex, surf zone and tropical region in the actual atmosphere. As I will show below, the results from high-resolution GCM simulations support this notion. The other interesting aspect of the satellite observations in Fig. 8 is the suggestion that the gradients are particularly concentrated along part of the periphery of the polar vortex, while at other points there may be an extrusion of vortex air into the surf zone. Again this kind of behavior is seen much more clearly in GCM results, and it appears that at least some of the vortex air in the extrusions can be irre- versibly mixed into the surf zone.

The mixing of air that produces the surf zone has important consequences for the large-scale dynamics of the middle atmosphere. Referring again to Fig. 7, one can see that the mixing of air in the surf zone involves an effective equatorward transport of PV. If the wind were geostrophically balanced, then it can be shown that the EPFD at any height and latitude is proportional to the meridional transport of PV across that latitude (e.g. Andrews et al., 1987). It is reasonable to suppose that in the real atmosphere the EPFD can be regarded as a rough sum of the geostrophic contribution (proportional to the PV transport) and a contribution from ageostrophic motions (notably inertia-gravity waves). The equatorward transport of PV in the surf zone of the winter stratosphere (Fig. 7) produces a westward EPFD, and so acts as a brake on the eastward jet set up by the radiative processes.


Fig. 8. The N,C1 mass mixing ratio at potential temperature 1150°K on December 17, 1991, as determined from the Improved Stratospheric and Mesospheric Sounder (ISAMS) instrument on the UARS satellite. Plotted in a NH polar stereographic projection. Redrafted from Ruth et al. (1994). The contours are labelled in part per billion by volume (ppbv). There are no observations in the black region around the

pole.

One of the critical issues in understanding the dynamics of the winter middle atmosphere is determining the relative contribution of gravity waves and breaking Rossby waves to the reduction in the radiatively-forced eastward vortex. GCMs will require an adequate representation of both processes to correctly simulate the zonal-mean flow and the transport of trace constituents in the middle atmosphere.

4.4. The zonal-mean circulation in the tropical stratosphere

The focus in this paper will be on the extratropical circulation, but I will make a few comments here on the rather different considerations that guide the study of the tropical middb: atmosphere. Since the zonal- mean zonal flow is nearly in thermal wind balance with the temperature field, a given wind perturbation is accompanied by a temperature perturbation that scales as the Coriolis parameter. Thus near the equator, the zonal-mean temperature changes (and hence induced mean residual vertical velocities) are small even for substantial wind changes. This fact, combined with the appearance of the Coriolis parameter in thefi* forcing term in the zonal momentum equation, leads to greatly reduced restoring effect of the diabatic circulation on the zonal-mean flow at low latitudes. This is reflected in the very different

circulation that develops in the tropical middle atmosphere. Near the equator the zonal-mean wind can evolve steadily in response to eddy forcing. This allows long-period oscillations to develop in the mean flow, notably the quasi-biennial oscillation (QBO) which dominates the flow in the tropical lower stratosphere and the semiannual oscillation (SAO) which is important throughout the upper stratosphere and mesosphere (e.g., Wallace, 1973; Hirota, 1980; Hamilton, 1987). The EPFD associated with vertically-propagating waves is thought to be crucial for the dynamics of these mean-flow oscillations (e.g. Holton and Lindzen, 1972; Plumb, 1977; Dunkerton, 1982; Sassi and Garcia, 1994).

5. THE SIMULATED MIDDLE ATMOSPHERIC

CIRCULATION

In this section I will review the actual performance of GCMs in light of the issues concerning the middle atmospheric circulation raised in Section 4. I will focus on results from the GFDL SKYHI GCM, but will try to place these results in a broader context of the performance of current models. I will consider the results for extratropical simulations first, and then briefly describe some aspects of the simulation of the tropical middle atmosphere. Most of the SKYHI

1608 K. Hamilton

results presented here have been discussed earlier in Hamilton et al. (1995) and Hamilton (1995a). The SKYHI integrations discussed in this paper do not inlcude a diurnal cycle and I will not deal with the issue of atmospheric tides here. Zwiers and Hamilton (1986), Tokioka and Yagai (1987) and Hunt (1991) have examined the solar tides appearing in GCM simulations, but this is an area in which much more work needs to be done.

5.1. The zonal-mean circulation in the extratropics

Historically, simulations of the stratospheric circulation in GCMs have been characterized by a winter polar temperature structure unrealistically close to radiative equilibrium. In the early papers of Kasahara and Sasamori (1974) and Manabe and Mahlman (1976) the winter polar temperature in the stratosphere is much too cold and the polar night jet much too strong both in the NH winter and (even more pronounced) in the SH winter. In recent years model results that show some improvement in this regard have been published. Such improvements apparently have resulted from increased spatial resolution and/or the inclusion of extra horizontal momentum sources meant to represent the effects of unresolved gravity waves (e.g. Mahlman and Umscheid, 1987; Rind et a/., 1988a,b; Boville, 1991).

Figure 9 presents the DJF climatology of zonal- mean zonal wind from observations (Fleming et al., 1988) and from simulations with each of the three different SKYHI horizontal resolutions. The results represent averages of 10 consecutive years of integration with the 3 x 3.6” model and 2 consecutive years for both the 2 x 2.4” and 1 x 1.2” models. In each case it is reasonable to assume that the results are essentially independent of the initialization employed (see Hamilton et al., 1995, for a discussion of the details of each model run). Even at 3 x 3.6” resolution the overall NH winter simulation is reasonably realistic, with a clear separation of the tropospheric and polar night eastward jets. The polar night jet is quite similar to the observations in the lower stratosphere. In the upper stratosphere the jet is of roughly the correct strength, but it is centered too far poleward. In the mesosphere the simulated jet closes off, in at least qualitative agreement with observations. The NH winter simulations at 2 x 2.4” and 1 x 1.2” resolution are fairly similar to that for the 3 x 3.6” model. As shown in Hamilton (1995b), the interannual variability in the NH winter stratospheric simulation is too great to regard the differences among the models in Fig. 9 as statistically significant.

The westward jet in the SKYHI simulated summer

hemisphere mesosphere shown in Fig. 9 is unrealistically intense, and does not completely close off in any of the simulations. This problem becomes pro- gressively less severe with increasing horizontal resolution, however. There is little interannual variability in the seasonal mean summertime circulation, so these differences among the models are almost certainly real. Connected with the deficiencies in the westward jet in the simulated mesosphere is a polar upper mesospheric temperature in the model that is unrealistically warm (by _ 35°C at 3 x 3.6” resolution and N 25°C at 1 x 1.2” resolution, at least when the observations of Fleming et al., 1988, are used for comparison).

Figure 10 shows the DJF climatology of force per unit mass acting on the zonal-mean zonal flow due to EPFD. The top-left panel shows the total EPFD in the 3 x 3.6” simulation, while the top right panel shows the EPFD calculated using monthly-mean fields (thus including only contributions from very low frequency eddies). The lower panels show the total EPFD for the two higher resolution models expressed as a difference from the 3 x 3.6” result. The EPFD in the extratropical summer (winter) hemisphere acts as an eastward (westward) force on the mean flow. This confirms the basic picture of eddy dynamics acting as a brake on the radiatively-forced zonal wind fields in both the summer and winter hemispheres. There is a clear difference between the hemispheres, however. In the summer hemisphere almost none of the EPFD is in the monthly mean fields, while much of the EPFD in the winter mesosphere is captured in the monthly mean (particularly in the stratosphere and lower mesosphere). This is consistent with the notion that the only important eddies for the momentum balance of the extratropical summer mesosphere are inertia- gravity waves, while the westward forcing of the mean flow in the winter hemisphere has contributions from quasi-stationary Rossby waves as well as from inertia- gravity waves. As horizontal resolution is increased, the eastward drag in the summer hemisphere mesosphere rises significantly. This is consistent with the reduced strength of the westward jet seen in the higher resolution simulations in Fig. 9. Miyahara et al. (1986) and Hayashi et al. (1989) showed that increasing horizontal resolution in the SKYHI model leads to an increase in the upward gravity wave momentum fluxes.

Figures 11 and 12 show the zonal wind and EPFD climatologies for JJA. The behavior of both the zonal wind and the EPFD as a function of resolution in the NH summer is quite similar to that seen earlier in SH summer. However, the SH winter simulation is considerably different from that in NH winter. In particular, the modelled SH polar night jet is much stronger


90s 80 33 EO 30 60 s

2Ox2.4’ SKYH

30 EO 30 60

903 60 30 EO 30

D.1

1.0

IO

loo

mb

Fig. 9. December-February climatology of the zonal-mean zonal wind from observations (top left), and from simulations with the 3 x 3.6” (top right), 2 x 2.4” (bottom left), and 1 x 1.2” (bottom right) versions of the SKYHI model. Contour labels are in m s-I and positive values (solid contours) represent eastward

winds. Observations from Fleming et al. (1988).

than observed at all three SKYHI resolutions. Accompanying the st.rong jet are JJA temperatures at the high southern latitudes that are much too cold, particularly in the upper stratosphere (see Hamilton et al., 1995). At 1 mb the simulated JJA temperatures near the South Pole are about 65,45, and 35°C colder than the observations in the 3 x 3.6”, 2 x2.4” and 1 x 1.2” models, respectively. The other obvious contrast with the NH winter is the near-zero contribution to the EPFD from the low frequency eddies (compare the upper right panels of Figs 10 and 12). The reduced influence of quasi-stationary waves is expected, given the much weaker topographic forcing in the SH. The westward driving of the mean flow by the eddies in the upper mesosphere does appear clearly in the SH

winter, and this forcing becomes stronger (and extends lower down) as the model resolution increases. The increasing horizontal resolution apparently results in stronger upward fluxes of gravity waves into the middle atmosphere and hence a more efficient brake on the eastward jet in the mesosphere. It is reasonable to speculate that at still higher spatial resolution the model might adequately represent both the gravity wave fluxes and the mean circulation in the SH winter. The fact in SKYHI that the winter middle atmospheric circulation is rather more successful in the NH than the SH, and the fact that it is less sensitive to model resolution in the NH can be attributed to the effects of large-scale Rossby waves (which should be well resolved even in the 3 x 3.6” model).

1610 K. Hamilton

’ T??f?&,O.Ol .I. mb 3Ox3.6’ 78.9 -

100 mb .--__ _.

1000 mb N

7ss- 2”x2.4”M

I\‘1 ‘I Ill wr , II II -1; :_ .-_.

INUS 3Ox3.6” o o1 mb

0.1 mb

1.0 mb

10 mb

3Ox3.6” TIME MEAN o,ol mb 7s.s- , , I , , , , , ,

I\ yz)---y

1 Ox1 .P”MINUS 3=.x3.6” o,ol mb

0.1 mb

1.0 mb

10 mb

100 mb

1000 mb Qd s sd s 36 s 3d N sd N 80 N

Fig. 10. Climatological December-February Eliassen-Palm flux divergence (EPFD) in the SKYHI simulations. The top left panel shows the total EPFD for the 3 x 3.6” model. The top right panel shows results for the low frequency eddies only (i.e. when only the monthly-mean fields are used in the calculation of EPFD). The lower left shows the total EPFD results for the 2 x 2.4” model minus those for the 3 x 3.6” model. The lower right panel shows the total EPFD results for the 1 x 1.2” model minus those for the 3 x 3.6” model. The contours in the top panels are for 2,4,6, 8, lo,20 m s-’ day-‘. In the lower panels the contours are for 1, 2, 3,4, 5, 8, 11, 14 m s-’ day-‘. In each case solid (dashed) contours represent

eastward (westward) forcing of the mean flow.

The results for SKYHI discussed in this section turn observed. Once his topographic gravity wave drag out to be somewhat atypical of current GCMs, at least parameterizaton is included, however, the simulated in the NH winter. Unlike SKYHI, most models now polar night jet strength is realistic, although the tend- being used have some parameterization of subgrid- ency for the vortex to be unrealistically confined near scale gravity wave drag. It appears that the drag par- the pole is evident in this simulation just as in the ameterizations included in these models are important SKYHI model. Indeed, Hamilton et al. (1995) noted in producing a reasonable simulation in the NH the very great similarities in NH winter climatology winter. The results of the NCAR CCM discussed by (including such aspects as eddy heat and momentum Boville (1991) are particularly relevant. He shows that transports) in the SKYHI model without par- without the topographic gravity wave drag parameterized gravity wave drag and the NCAR CCM ameterization, the simulated zonal-mean circulation with the topographic drag included. In the SH, of in the NH winter is very much more intense than course, the parameterized tropospheric drag in the


12.8

4.0

Qm 60 30 EO 30 60 QON

79.8

88.3

56.6

45.4

rnb 36.1

23.0

20.9

12.8

40

Qns 60 30 EO 30

Fig. 11. As in Fig. 9, but for June-August.

NCAR model has only minor influence and the SH winter simulation is characterized by a much too cold pole. Garcia and Boville (1994) and Boville (1995) interpret this as evidence that a parametrization of subgrid-scale gravity wave effects associated with nonstationary waves needs to be incorporated into the model. Hamilton (1995c) showed that the SKYHI model can produce quite a good simulation of the circulation throughout the middle atmosphere during SH winter, if a particular zonally-symmetric distribution of zonal drag is imposed in the extratropical mesosphere.

5.2. Large-scale stationary eddies in the winter extratropics

The left hand panels in Fig. 13 show the long term mean NH 10 mb, 30 mb and 50 mb DJF mean geo-

potential heights from the 3 x 3.6” SKYHI integration. The corresponding right hand panels show the same climatological fields computed from the Free University of Berlin subjective NH analyses based on radiosonde data (see Hamilton et al., 1995, for details). The same basic features of the wave field are apparent in both model and observations. The presence of both zonal wave 1 and 2 components is seen as an elongation of the polar vortex roughly along the 9O”E-9O”W axis, and a displacement of the vortex center off the pole roughly along the Greenwich meridian. The phase of the whole pattern tilts westward with height, so that in the observations the Aleu- tian high is centered near 15O”W at 50 mb and near 175”E at 10 mb. The SKYHI simulated stationary wave phase tilt is even more pronounced, with the Aleutian high at 10 mb centered near 160”E. The

1612 K. Hamilton

JO.1 mb ““-“--L/I

78.8- 3Ox3.6“ TIME MEAN .,o, mb

I - / I JlOmb :: r7 I

100 mb

1000 mb N

2”x2.4”MINUS 3Ox3.6’ o,o, mb 1’x1.2”MINUS 3Ox3.6’ o,ol mb

0.1 mb

1.0 mb

10 mb

100 mb 100 mb

1000 mb 90 s 80 s 30 s 30 N 60 N 00 N go s 60 s 30 s 30 N BON SON

Fig. 12. As in Fig. 10, but for June-August.

magnitude of the geopotential gradients seen at 50 mb and 30 mb in the model agree reasonably well with those observed. However, a comparison of the time mean geopotential maps at 10 mb shows signs of the model bias noted earlier, i.e. a tendency to have a polar vortex that is unrealistically confined to high latitudes. The geopotential gradients in the model vortex at high latitudes are clearly stronger than those indicated in the observations.

The geopotential analyses from the Free University of Berlin are limited by the availability of balloon- borne radiosonde data and so extend only up to 10 mb. Hamilton et al. (1995) show a comparison extending up to 0.01 mb between the zonal wave 1 and 2 components of January-mean temperature in NH winter simulated in the 3 x 3.6” model and those derived from satellite radiometer data by Barnett and Corney (1985). Again there is reasonable overall

agreement, but the tendency in the model for the wave amplitudes to be concentrated unrealistically near the pole is evident.

It is important to appreciate that the vertical modulation of the large-scale extratropical stationary waves revealed in Fig. 13 actually implies that the stationary waves must be strongly dissipated. This is shown clearly for the model results in Fig. 14 which presents the vertical EP flux associated with the January stationary waves in the 3 x 3.6” model at levels from 62 mb (-20 km) to 1 mb (-50 km). There is an order-of-magnitude drop in the flux between these two levels. The interpretation of this is not completely unambiguous, since such a drop could be produced by a refraction of EP flux from the vertical to the horizontal direction, but the very strong decline (and the fact that there is a significant divergence in the total EP flux associated with the stationary waves in


ZlO control ZlO observed

230 control 230 obSeNed

Z50 observed 250 control

Fig. 13. Long tlerm-mean DJF geopotential heights at 10 mb (top), 30 mb (middle) and 50 mb (bottom) from the 3 x 3.6” SKYHI simulation (left) and from observations (right). The contour labels are in 100’s of m, and the contour interval is 200 m. The NH is shown in an orthonormal projection. Dashed circles

are placed at 30” and 60”.

1614 K. Hamilton

O.l6r------ 0 I/\’ 8 1 ’

_ 0.12 - o? 9 n $ 0.10 - g

2 0.06

4 jj 0.06 - ._ % ’ 0.04 -

0.02 -

0.00 I 0 10 20 30 40 50 60 70 60 90

Latitude Fig. 14. The vertical component of Eliassen-Palm flux (normalized by the radius of the earth) associated with the January-mean fields in the 3 x 3.6” SKYHI simulation. Results presented for several individual model levels in the stratosphere. Each curve is labelled by the pressure of the level in mb. Reproduced from

Hamilton (1994).

the winter stratosphere, see Fig. 10) imply that there is indeed a strong effective dissipation of the stationary waves. This may be an indication of irreversible nonlinear breaking of the large-scale Rossby waves in the middle atmosphere. It is noteworthy that the frac- tional drop in flux with height in Fig. 14 is smallest at high latitudes where the presence of the vortex edge provides a strong restoring force, and greatest in the midlatitudes (where potential vorticity gradients are presumably weakened by the horizontal mixing in the surf zone).

Hamilton (199%) shows some results for stationary waves in the winter months of the SH in the 3 x 3.6” SKYHI simulation. In the standard run, the zonal- mean circulation is so unrealistic that comparison of the eddy fields with observations is not meaningful. However, when an extra mesospheric drag is imposed to produce realistic zonal-mean flow, the stationary wave field in the middle atmosphere is reasonably well simulated by the model.

5.3. Evidence for a surf zone in winter midlatitudes

Examination of the geopotential (or the temperature or horizontal wind fields) in the extratropical winter stratosphere clearly shows the intense polar vortex, but the detailed structure of the winter hemi- spheric circulation is more fully revealed by inspecting either potential vorticity or a Lagrangian marker of motions such as long-lived trace constituents or clus-

ters of air parcel trajectories. In fact, the separation of the winter hemisphere middle atmosphere into a polar vortex, a surf zone and a tropical region is very clear in the tracer fields simulated by the SKYHI model. Figure 15 shows the instantaneous N,O field on the 850°K potential temperature surface (- 30 km) on a day in NH midwinter in the 1 x 1.2” SKYHI model simulation (see Hamilton and Mahlman, 1988, for a description of the implementation of the N20 tracer within SKYHI). The low values in the vortex, the very strong gradients along the vortex edge, the more homogeneous values in the surf zone, and another region of strong gradients in the subtropics are all very apparent in Fig. 15. The presence of very strong gradients at the vortex edge is consistent with McIntyre and Palmer’s (1984) notion of a surf zone generated by irreversible mixing associated with nonlinear quasi-stationary Rossby waves. Eluszkjewicz et al. (1995) discuss in more detail aspects of the N20 simulation in the 1 x 1.2” SKYHI model simulation and show air parcel trajectories computed from the model winds. Their results also demonstrate that the rate of horizontal mixing of air is very much enhanced in the midlattude region outside the vortex.

One exciting possibility with the detailed data available in a GCM is the investigation of the three-dimensional structure of the vortex and surf zone. This is an area that has not been explored deeply, but Hamilton et al. (1994) present a preliminary study using the N,O

Comprehensive meteorological modelling of the middle atmosphere: a tutorial review

4’

1615

-QO’W

Fig. 15. Instantaneous NzO mixing ratio on a day in mid-January as simulated in the 1 x 1.2” SKYHI model control run. Plotted in a NH polar stereographic projection. Results have been slightly smoothed

in the horizontal. The contour interval is 20 ppbv and the 100 ppbv contour is plotted as a heavy line.

simulation in the 1 x 1.2” SKYHI model. In particular, they examined some three-dimensional visual- izations of a surface surrounding low N,O air in NH midwinter. These slnowed the outline of the polar vortex quite clearly as well as the extrusions of vortex air into the midlattudes. The extrusions clearly have some considerable vertical coherence, but may not extend through the entire stratosphere. Hamilton et al. (1994) noted in particular that some of the lower stratospheric extrusions iappear to be quite shallow.

5.4. Model results in the tropics

Here I want to describe very briefly some results for the tropical middle atmospheric circulation as simulated by GCMs, and to introduce the reader to the important published papers on this issue. The zonal- mean circulation near the equatorial stratopause in the real atmosphere is dominated by the SAO, with strong eastward winds peaking a few weeks after the equinoxes and wesl:ward winds strongest after the sol- stices (e.g. Hirota, ‘1980; Hamilton, 1987). GCMs have had considerable success in simulating the stratopause SAO. Hamilton and Mahlman (1988) and Hamilton et al. (1995) examined the SAO in the SKYHI model, while Sassi et al. (1993) and Gray and Jackson (1995) studied the SAO as simulated by the NCAR CCM and the UGAMP middle atmosphere model, respectively. In all cases the prevailing zonal-mean winds show a clear SAO with roughly the observed phase.

The amplitude of the oscillation tends to be somewhat underestimated (say by -30%) in all these models. The strongest equatorial eastward winds in each of the models is weaker than observed. The strength of the westward phase may also be somewhat underestimated. As noted above, Hamilton (1995~) examined the 3 x 3.6” resolution SKYHI results in a version with an imposed zonal momentum forcing in the extratropics of the SH through the winter season designed to alleviate the cold pole problem so apparent in the standard model. The addition of this extratropical momentum source increases the strength of the westward phase of the SAO at the equator, consistent with the notion that the westward phase may be driven largely by the advection of mean momentum by the diabatically-forced cross-equatorial flow (e.g. Holton and Wehrbein, 1980; Mahlman and Sinclair, 1980). The issue of how the eastward phase of SAO simulation might be affected by a parametrization of gravity wave EPFD acting locally in the tropics is addressed by Sassi and Garcia (1994).

A second peak in the amplitude of the SAO in zonal wind is observed near the equatorial mesopause (Hirota, 1978; Hamilton, 1982; Paolo and Avery, 1993) with the opposite phase of that seen at the stratopause. Dunkerton (1982) suggested a plausible explanation for the mesopause SAO in terms of driving by EPFD associated with dissipating gravity waves. Such waves will have their phase speed spec-

1616 K. Hamilton

trum filtered during their passage through the middle atmosphere. The expected result is that when the stratopause SAO is near its eastward extreme there should be a net westward bias in the gravity wave spectrum reaching the mesopause. Hamilton and Mahlman (1988) and Hamilton et al. (1995) show that in the SKYHI model there is a mesopause SAO with observed phase, but with perhaps 50% of the observed amplitude. Again it is tempting to speculate that the GCM has produced a mesopause oscillation by virtue of its resolved gravity waves and to attribute the unrealistically weak amplitude to the lack of unresolved gravity waves. This is an aspect of upper mesospheric/lower thermospheric dynamics that is ripe for further observational and modelling study.

GCMs have been much less successful in simulating the QBO than the SAO. Typically model simulations in the tropical lower and middle stratosphere are characterized by a relatively weak (- 10 m ss’) westward flow at low latitudes which displays little interannual variability. When examined closely, the equatorial wind fields in at least some models are seen to undergo an oscillation of roughly biennial period (Cariolle et al., 1993; Hamilton et al., 1995), but with a peak-to- peak amplitude perhaps - 10% that of the observed QBO. More promising, perhaps, are some experi-

ments that show that GCMs altered in various ways (or with perturbed initial conditions in the tropical stratosphere) can produce mean flow evolution with some similarities to that seen in the observed QBO (Mahlman and Sinclair, 1980; Hamilton and Yuan, 1992; Hamilton, 1995~; Takahashi and Shiobara, 1995).

6. VERTICALLY-PROPAGATING HIGH-FREQUENCY

WAVES IN THE MIDDLE ATMOSPHERE OF GCMs

As noted in the previous section, there seems to be an important contribution to the zonal-mean zonal momentum budget in the simulated SKYHI middle atmosphere from explicitly-resolved gravity waves. There is also reason to suspect that some of the deficiencies in model simulations of the middle atmospheric circulation may be due to the inability to resolve small-scale gravity waves. In this section I want to look in more detail at the nature of the explicitly-simulated gravity wave field in GCMs and consider the problem of parameterization of effects of subgrid scale gravity waves. These are subjects of very active investigation now, and here I will simply try to introduce some preliminary findings and touch on the important issues that need to be addressed.

44.5’N, 5.4”E June 6-15

80

60

0 1 2 3 4 5 6 7 8 9 10 TIME (days)

Fig. 16. The time series of zonal wind simulated at the gridpoint at (445”N, 5.4”E) during the control run of the 1 x 1.2” model. The series are shown for each of the top 6 levels and the lo-day mean is subtracted from each series. The scale at the left gives the approximate height of each level The arrow on the right

hand side represents 40 m SK’.


6.1. Wind and temperature manifestations of the gravity waveJield

Figure 16 displays the time series of zonal wind at a gridpoint at 45”N from five days of a 1 x 1.2” SKYHI integration sampled at 1 h intervals. Results are presented at each of the top six model levels. One could imagine plotting radar (or Doppler lidar) observations of the wind in a similar format. There is an impressive amount of high-frequency (periods of hours to days) variability evident in this time series. One obvious failing of the model. is at very high frequencies. While radar winds are observed to vary significantly on timescales substantially less than an hour, the SKYHI winds have little variance at periods less than - 2 h. This is most reasonably attributed to the limited horizontal resolution in the model, and in fact the cutoff frequency for substantial variability is dependent on the model resolution employed (see Hayashi et al., 1989; HamiltlDn, 1995d). A general tendency for downward phase propagation is apparent in Fig. 16, consistent with the interpretation of these fluctuations as evidence of gravity waves with upward energy propagation. The dominant vertical wavelengths appear to be -l&20 km (see also Manzini and Hamilton, 1993). It would be of great interest to compare the statistics of the horizontal wind field fluctuations in the model with those observed by radars. Unfortunately, the wind measurements from radars are generally available only near and above the mesopause, while the SIKYHI model results end near the mesopause. The interest in comparing model results with radar data is one of the prime motivations for extending GCMs up into the lower thermosphere.

One valuable source of wind (and temperature) observations in the stratosphere is the historical archive of rocketsonde profiles, which typically cover the 30-60 km altitude range. Unlike radar measurements, however, these observations are almost always spaced too far in time for any useful information to be directly determmed about time scales for gravity waves. Fortunately, the simultaneous high vertical resolution observations of the temperature and horizontal wind provided by rocketsondes can actually yield a great deal of insight into the dominant wave characteristics (Hirota and Niki, 1985; Hamilton, 1991; Eckermann et al., 1995). In particular, the horizontal wind vectors in an individual sounding typically display a clockwise rotation with height (in the NH, opposite in l:he SH) which is consistent with inertia-gravity waves with upward energy propagation. The hodographs in these soundings generally have a roughly elliptical polarization. The statistics of the wind variations can be expressed in terms of the

amplitude, polarization, eccentricity and orientation of the hodographs. Hamilton (1989, 1993) describes some comparisons between the statistics of the gravity wave field in the SKYHI model and in a large number of rocket profiles taken over a lo-yr period at several stations in locations extending from the tropics to near the Arctic circle. The comparisons were generally quite favorable, suggesting that the model is producing a random gravity wave field in the 30-60 km altitude range with roughly the correct energy, dominant periods and directions of propagation. The greatest differences between the model and observations occurs at high latitudes (poleward of about 60”) where the model significantly underestimates (by a factor -2) the amplitude of the wind and temperature fluctuations seen in the rocket soundings.

The development of the Rayleigh lidar technique over the last decade has provided a method of observ- ing the high-frequency variations in middle atmospheric temperature. Particularly valuable for present purposes are the long term observations that have been made with the lidar at Obervatoire de Haute Provence (OHP, 44”N, 6”E). Wilson et al. (199la,b) present results for the temperature field between about 30 and 75 km over several years. Wilson et al. (1991a) show some typical height-time cross-sections for individual nights. For comparison, Fig. 17a presents a section of temperature fluctuations at the model gridpoint nearest OHP during 5 days of simulation with the 1 x 1.2” SKYHI model. This resembles the typical OHP lidar results in many respects, although the observations do clearly show more variability at high frequencies (periods < 2-3 h). Wilson et al. (1991 b) computed a measure of the energy density per unit mass in these fluctuations E = 0.5 (g/N)‘( T/To)*), where N is the Brunt-Vaisala frequency, TO and T are the mean and fluctuating components of the temperature, and the angle brackets denote an average over many individual profiles. Wilson et al. present averages of E over the height ranges 30-45 km, 45-60 km and 60-75 km for each month of the year. Figure 17b compares their results for June with a similar analysis of 1 x 1.2” SKYHI model results. The agreement is quite good, again suggesting that the overall amplitude of the middle atmospheric gravity wave field is reasonably well simulated in the model.

While there is generally good agreement between the high frequency fluctuations seen in observations and the SKYHI model, there is an important limi- tation to such comparisons. It is easy to show that for a field of linear gravity waves with phase velocities aligned in the same direction, the horizontal spatial spectrum of the vertical eddy flux of momentum u’w’ will be considerably shallower than the spectrum of

1618 K. Hamilton

a SKYHI “OHP” Temperature

36.1

26.0

, u I

June 9 June 10 June 11 June 12

b June Mean Gravity Wave Activity Near 44ON, 6OE

OHP LMar Observations “(I t- SKY”,

I 0 20 40 so 80

Enerpy Per Unit Mass (J-W’)

Fig. 17. (a) Temperature anomalies at the gridpoint corresponding to Observatoire de Haute Provence (OHP) over 5 days of simulation with the 1 x 1.2” SKYHI model. The zero contour is shown along with regions where the temperature anomaly is > 10°C (dark shading) or < - 10°C (light shading). (b) Comparison of estimated gravity wave energy density based on OHP lidar observations and the 1”

SKYHI simulation.

kinetic energy. Thus the energetic waves that dominate the observations of wind and temperature fluctuations may not represent the most important part of the spectrum in terms of the effects on the mean flow. This is a point that will be pursued in the next section.

6.2. Horizontal spectrum of simulated gravity waves A very important aspect of the gravity wave field in

the middle atmosphere is its horizontal spectrum. This

is again an example of an issue that is extremely difficult to examine with real observational data, but one that can be studied readily in a GCM. Hamilton and Mahlman (1988), Hayashi et al. (1989), Manzini and Hamilton (1993) and Hamilton (1993, 199%) have all addressed aspects of this issue. Here I want to follow Hamilton (1993) in considering the extratropical spectrum in the SKYHI model. Figure 18 shows the cospectrum of zonal wind and vertical velocity resolved into zonal wavenumbers around a lati-


Vertical Eddy Momentum Flux near 45”N 0.0308 mb June/July

““I

,I,,, 10 100

Zonal Wavenumber Fig. 18. Zonal wavenumber cospectrum of u’and o’ near 45”N in simulations in June and early July for the 1 x 1.2” SKYHI model (solid) and the 2 x 2.4” model (dashed). Results in each case represent averages over a large number of instantaneous spectra and have been smoothed somewhat in zonal wavenumber.

tude circle near 45”N in the mesosphere, computed from month-long segments of simulations with both the 2 x 2.4” and 1 x 1.2” SKYHI models. Results here are for NH summer and in each case represent averages over a large number of spectra computed from individual snapshots. The plotting convention is such that the positive values seen at all wavenumbers means that each wavenumber is making a negative contribution to u’w’ and hence an eastward forcing of the mean flow where the waves are dissipated. Note that for these average spectra, the sign of the flux is the same at all wavenumbers. This behavior is an impressive illustration of the efficiency of filtering effects of the mean flow discussed above in Section 4.2.

A very striking aspect of each of the momentum flux spectra in Fig. 18 is its shallowness. After wavenumber - 30 both the 2 x 2.4” and 1” x 1.2” results follow a power law k-8, where k is the zonal wavenumber and /I - 2/3. There is some deviation from this near the highest resolved wavenumbers at both model resolutions, but the overall effect is evident, with the higher resolution model results simply extending those of the coarser model out to higher wavenumbers. As noted in Hamilton (1993), the km@ spectrum is very shallow

in the sense that it is actually not integrable over all wavenumbers. In practice, of course, this spectrum must drop off more rapidly beyond some cutoff wavenumber. However, assuming that the k-fl, behavior does continue significantly beyond the highest resolvable wavenumber, the implication is that much of the total eddy momentum flux cannot be explicitly resolved. Hamilton (199%) shows that the situation for the spectrum of waves in the winter midlatitudes is somewhat more complicated (in particular, the momentum fluxes do not follow a simple power law), but the results do support the notion that the simulated spectrum of momentum flux is quite shallow. All these results are consistent with the idea introduced earlier that GCMs will need to include a significant drag in the mesosphere associated with the effects of unresolvably small gravity waves in order to obtain a realistic simulation of the middle atmospheric general circulation.

6.3. Heating of the atmosphere by turbulent dissipation

An aspect of the gravity wave field of particular interest in the upper mesosphere and lower thermosphere is the atmospheric heating expected to result

1620 K. Hamilton

from mechanical dissipation of the waves. Here I want observations. In particular Lubken et al. find dis- to briefly review some relevant observations and some sipation rates of the order of 10 mW kgg’ at 80 km preliminary model results (reported in Hamilton, and suggest that this value falls off rapidly to - 1 mW 1995a). kg-’ in the lower mesosphere.

The growth of the wind and temperature amplitudes of the waves accompanying the decreasing mean density with height eventually should lead to some kind of nonlinear breaking and the consequent generation of turbulence. Within the turbulent regime there is presumably a cascade to smaller scale motions that are ultimately dissipated by molecular viscosity and heat diffusion. The turbulent dissipation of kinetic energy in the mesosphere and lower thermosphere has been estimated using various observational tech- niques. Hocking (1985) reviewed a number of published estimates for the 8&120 km range. For winter high latitudes his review suggests a mean value of the order of 100 mW kg-’ for the turbulent dissipation at 80 km (although with a great deal of variability among individual profiles). More recently Lubken et al.

(1993) have used a new technique to estimate energy dissipation in the 6@100 km height range. Analysis of data taken during a number of rocket flights in mid and late winter at Andoya (69”N) suggested rather lower values for the dissipation rate than the earlier

Given the limited spatial resolution in a GCM, an explicit simulation of the isotropic turbulence regime is obviously not possible. However, subgrid scale mixing parameterizations are included in an attempt to account for the effects of unresolved scales of motion on the resolved flow. The energy lost by the resolved flow to the subgrid scale parameterizations is assumed to cascade to ever smaller scales and end up as isotropic turbulence (which is ultimately dissipated by molecular viscosity). Thus it may be reasonable to compare the model energy dissipation due to parameterized subgrid scale processes (i.e. the term - F,u - Fp in equation 2.4) with the observed estimates of small scale turbulent dissipation. There are caveats to be noted, however. One concerns time scales. If the cascade from grid scale (which may be - 100’s of km) to the dissipation range in the real atmosphere takes sufficiently long, then the actual dissipation may be significantly nonlocal. The other complication is that the subgrid scale parameterizations remove not only kinetic energy, but also potential energy (this can

0.01

0.1

100 - 0 5 10 15 20

Dissipation Rate (mWkg) Fig. 19. Dissipation of kinetic energy by the subgrid scale parameterizations in the 1 x 1.2” SKYHI model. Results are for February means and averages around the 69S”N latitude circle. Values are plotted up to

the second highest model level (0.0308 mb).


occur particularly in the application of dry convective is convective adjustment or where the resolved-scale adjustment). This energy loss from the resolved scales Richardson number fails below some threshold). The might also ultimately end up as kinetic energy of iso- treatment of the problem beyond that point in the tropic turbulence. In the present paper only the direct Rind et al. scheme is rather simple, however, with an dissipation of resolved scale kinetic energy by the assumption that the subgrid-scale waves has a unique subgrid scale mixing parameterizations will be wavenumber and phase speed, and can be treated by computed. steady-state WKB approximations (similar to equa-

Figure 19 shows this dissipation rate in the 1 x 1.2” tions 4.12 and 4.13) with a further assumption that SKYHI model integration averaged (from daily nonlinear processes lead to a saturation of wave instantaneous samples) over the month of February amplitudes (following Lindzen, 198 1). and averaged around a latitude circle near 70”N. The The other set of approaches that have been results are plotted only up to the second highest model advanced begin with the assumption that a sta- level (0.0308 mb) since the artificial eddy damping tistically-steady spectrum of waves is emerging from imposed at the top level distorts the results there. the troposphere at all times (the spectrum can, of The dissipation rate rises rapidly with height in the course, be specified as a function of location and time mesosphere, in agreement with the observations of of year). The next step is to compute the EPFD that Lubken et al. When extrapolated in a straightforward results from the propagation and dissipation of these manner to 0.01 mb (-80 km) results would imply waves. Here again there are some quite different pro- dissipation rates -20 mW kg-‘, a figure which lies posals that have been published. One simple approach between the observational estimates of Lubken et al. is to follow Lindzen (198 1) and compute the effects of (1993) and Hocking (1985) for that level. This suggests a number of discreet waves individually (i.e. ignoring that the overall strength of the gravity wave energy any nonlinear interaction among the waves). This has fluxes in the extratropical mesosphere is captured in a been employed in the simple middle atmosphere models fairly realistic manner by the GCM. The 20 mW kg-’ of Holton (1983) and Garcia and Boville (1994) and value corresponds ‘to a heating rate of -2°C day-‘, is now being adopted in the NCAR CCM. Fritts and showing that the turbulent heating ought to be VanZandt (1993) and Fritts and Lu (1993) have for- included explicitly in the thermodynamic equation of mulated a more sophisticated treatment involving a GCMs that extend up to the mesopause. continuous spectrum of waves, but again essentially

assuming that each part of the spectrum attains a

6.4. Parameterization of the gravity wave effects on the nonlinear saturation independently of the rest of the

mean flow spectrum. By contrast, Hines (1991, 1996) presents a theory of the upward propagation of a spectrum of

I have shown a number of results that suggest that waves that considers the mutual interaction among the performance of GCMs in the middle atmosphere the waves as the key to their propagation and dissi- could be improved by adding an extra drag on the pation. In particular, Hines has produced a formalism zonal mean flow in the mesosphere. This drag is most in which the dissipation of waves (and hence their plausibly associate’d with unresolved gravity waves EPFD) is controlled in large part by a Doppler spread- generated by various sources in the real troposphere. ing induced by the winds of the wave spectrum itself. The question of how to parameterize effects of such It would seem that the Hines approach would be waves in GCMs, a.nd particularly the effects of the appropriate if the gravity wave field were very steady, nonstationary component of the wave field, is now while the approaches based on saturation of indi- being very actively investigated. Here I will only give vidual waves would be relevant to a case where the a very broad overview of the current state of this wave field was extremely transient, with pseudo- problem. monochromatic waves dominating each local region

Some widely-varying approaches to the practical for significant times. The range of validity of each parameterization problem have been proposed thus of these approaches needs to be understood more far in the literature. The differences among various completely, and these idealized theories have to be proposals relate to both the specification of sources of adapted to produce results relevant to the complicated subgrid-scale waves and the treatment of their vertical situations encountered in GCMs. It seems fair to con- propagation and dissipation. At one extreme Rind et clude that it is unlikely that a credible a priori par- al. (1988a) have a source of waves that depends at ameterization will be developed in the near future, each gridpoint and timestep on the local tropospheric and practical parameterization schemes in GMs will conditions (so that subgrid-scale gravity waves are have to have their input parameters tuned to produce ‘launched’ at the tropopause at gridpoints where there realistic middle atmospheric simulations.

1622 K. Hamilton

7. CONCLUSION

I have tried to summarize the important issues in comprehensive model formulation and analysis as they particularly relate to the middle atmosphere and lower thermosphere. The description of the basic elements of GCMs showed that significant problems remain unsolved. Some of these are common to all climate models (notably the parameterization of convection and cloudiness) while others are of concern only for models extending at least into the mesosphere (notably the non-LTE radiative effects at low atmospheric density).

I presented a very condensed summary of the theoretical background that underlies current understanding of the dynamics of the middle atmosphere, including such subjects as the EP flux, transformed Eulerian mean-flow diagnostics, gravity wave drag and the notion of breaking Rossby waves and the middle atmospheric surf zone. I have tried to emphasize the symbiotic interaction between the theoretical ideas and GCM experiments. Theory can guide the analysis of GCMs (and hopefully indicate where to make model improvements), but the GCM results can also help confirm and sharpen theoretical ideas. In particular some theoretical notions can only be examined thoroughly in a comprehensive model for which the fields (including vertical velocity and diabatic heating rates) are known exactly at high spatial and temporal resolution. I have shown that results from GCMs have helped support the importance of both gravity wave drag and the mixing processes in the surf zone for determining the large-scale wind and temperature structure in the middle atmosphere.

From the point of view of modelling the upper mesosphere and lower thermosphere, perhaps the most significant aspect of comprehensive models is their ability to produce a self-consistent field of upward-propagating gravity waves. In Section 6 I showed just how rich the spectrum of explicitly-simulated gravity waves can be. There are a variety of research directions that can take advantage of the

availability of models with explicitly-simulated gravity waves. One is the study of the effects of the gravity waves on the diurnal tides. There has been considerable work on this subject using simple parameterizations to represent the gravity wave effects (e.g. Walterscheid, 1981; Walterscheid et al., 1986; Forbes et al., 1991; Lu and Fritts, 1993; Hagan et al., 1995). Such studies suffer from the arbitrary assumptions in the formulation of the gravity wave parameterization. Diagnosis of the results from an integration with the diurnal cycle in a full GCM could shed light on the tidal-gravity wave interaction problem. The explicitly-resolved field will likely omit some important small-scale waves, of course. However, just as in the case of the zonal-mean circulation, it may be possible to use the results of GCM integrations at several different spatial resolutions to make inferences concerning the gravity wave effects on tides expected in a model with extremely high resolution model (or indeed in the real atmosphere). Another important application of GCM results is in the diagnosis of the significant excitation mechanisms for gravity waves in the atmosphere (e.g. Manzini and Hamilton, 1993).

This survey concluded with a brief discussion of the problem of parameterizing the effects of unresolved gravity waves in GCMs. Along with the treatment of moist convection and cloudiness, the gravity wave parameterization problem is one of the most significant issues now confronting climate modellers.

The application of GCMs to the stratosphere, mesosphere and lower thermosphere is still an emerging area of study. I hope I have made clear the chal- lenges remaining in both model development and analysis, as well and the opportunities for increasing our understanding of atmospheric dynamics through the study of model results.

Acknowledgement-The author wishes to thank R. J. Wilson for his assistance with the SKYHI model integrations discussed here. J. D. Mahlman, A. Hall, A. Klonecki, R. Orris and the official reviewers provided helpful comments on the manuscript.

Andrews D. G. and McIntyre M. E.

Andrews D. G., Mahlman J. D. and Sinclair R. W.

Andrews D. G., Holton J. R. and Leovy C. B.

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Comprehensive meteorological modelling of the middle atmosphere: a tutorial review