Meteo 465

Middle atmospheric dynamics

 

 

You can augment your reading with Holton, Chapters 10 and 12.

 

While it is correct to state that the general atmospheric circulation is driven by the differential absorption of solar energy at the surface, this statement may be only partially correct for some regions, such as the stratosphere.  For the stratosphere, eddies are as fundamental to the circulation as the differential solar heating.

 

In discussing the general circulation, we will be looking for those processes that maintain the mean zonal, and meridonal circulations.  We will see that these are coupled and that waves have a lot to do with it.

 

Without eddies,

·       the zonal mean temperature would match, with a ~10-20 day lag, the radiative equilibrium temperature and would vary annually; 

·       the flow would be only the zonal mean flow, which is determined by the meridional temperature gradient and the thermal wind balance;

·       there would be no stratosphere troposphere exchange;

 

Heating and cooling patterns result from eddies driving the stratosphere away from radiative equilibrium.

 

Important quantities:

 

1. Potential temperature (units: K). 

q = T (po/p)R/cp  =  T (1000/p)0.286

 

Potential temperature is conserved when there are no diabatic effects.  An air parcel will tend to move on an isentropic surface, or surface of constant potential temperature.

 

2. (Ertel’s) Potential vorticity (units: K kg-1 m2 s-1).

 

PV  =  (z + f) (-g q /p)

 

where z = k·(Ñ x u) is the relative vorticity, which can be either shear vorticity or curvature vorticity.

 

PV is conserved in an adiabatic, frictionless flow.

The figure give the PV in the stratosphere relative to the troposphere.

 

3.  Geopotential height:

 

F = ò g dz

 

We need also consider geostrophic winds, which are the balance of Coriolis and pressure gradient forces.

 

vg = - (1/f) F/x

 

ug = - (1/f) F/y

 

where F = ò g dz, the geopotential height, R = gas constant, H = scale height, and f = 2W sinj ~ 10-4 s-1 @ 45oN, the Coreolis parameter.

 

In the analysis to follow, we will assume that the motions are approximately geostrophic, or quasi-geostrophic.

 

Now,                                                                                                  

 

From these equations, we can derive the thermal wind equations:

 

  and  

 

where the “p” subscript implies that the differentiation occurs while the pressure is constant.

 

Let’s look at the zonal mean circulation first and then we will move on to the mean meridional circulation.  We will use the log-pressure coordinate:

 

z = -H ln(p/po),

 

where po is taken to be 1000 hPa.

 

1.  Start with the equations of motion:

conservation of momentum:

 

 

 

where the first term is the total derivative of u and v, , the second term is the Coriolis forcing, the third term is the pressure gradient forcing, and X and Y are zonal and meridional components of drag forcing due to small eddies.

 

hydrostatic equation:

 

 

where H is the scale height.

 

conservation of mass:

 

 

and thermodynamic energy equation (conservation of energy):

 

 

where  , and J is the diabatic heating rate.

 

 

 

 

 

2.  Find the zonal means, which will be needed to examine the vertical and meridional flows.  To do that, we must first write all the variables as the sum of a mean and a perturbed (or eddy) part.  Thus,

 

u = ū + u’.

 

The Eulearian means are found by taking the zonal averages for a fixed latitude, altitude, and time

 

 

 

 

 

Assume that the flow does wander too far in the meridional direction, so that we can use the beat plane approximation.

 

, where fo = 2W sinfo and  b = 2 W a-1 cosfo and A is Earth’s radius = 6400 km.

 

 

The result of all these assumptions is the equations:

 

zonal mean momentum equation:

 

 

change in the mean zonal momentum with time           Coriolis forcing           mean eddy momentum flux divergence            mean zonal eddy drag

 

thermodynamic energy equation:

 

T change with time      adiabatic cooling         mean eddy heat flux divergence          diabatic effects

 

 

 

 

where is N is the buoyancy frequency, the atmosphere’s natural resonance frequency for gravity-forced oscillations.  We have neglected advection by ageostrophic mean meridional circulation and by vertical eddy flux divergences

 

The meridional momentum equation can be found by assuming geostrophic balance: 

 

which when combined with the hydrostatic relationship, gives the thermal wind relation:

The ageostrophic mean meridional circulation is constrained by the thermal wind equation: the relationship between the zonal mean wind and the potential temperature distribution.

 

Without a mean meridional circulation, , the eddy momentum flux divergence and the eddy heat flux divergence would tend to change the mean zonal wind and temperature fields so that thermal wind balance would be destroyed.

 

But small departures of the mean zonal wind from geostrophic balance cause a mean meridional circulation, thus maintaining the thermal wind balance.  So a balance is established in the meridional direction so that

 

the Coriolis force  ~ the divergence of eddy momentum fluxes

adiabatic cooling         ~ diabatic heating and convergence of eddy heat fluxes

 

Look at the figure on the atmosphere in radiative equilibrium versus reality.  The temperature difference between the summer and winter poles in the 30-60 km region is less than expected from radiative equilibrium.  Above, 60 km, the gradient is even backwards.  To understand these differences, the Eulerian Mean Circulation does not account for the tendancy of eddy heat flux convergence and adiabatic cooling to cancel each other out, with the diabatic heating being a small factor.

 

Thus, an air parcel can rise only if its potential temperature is increased by diabatic heating.  It is this small diabatic heating gives rise to a residual meridional circulation, which in turn determines the mean meridional mass flow. 

 

We can define the residual circulation as follows:

 

Now adiabatic motions and eddy thermal flux divergence and convergence are accounted for without assuming that they drive the circulation.

 

 

So, the zonal mean momentum equation and the thermodynamic energy equations have the form:

 

 

 

 

where F is the Eliassen-Palm (EP) flux, which arises from large-scale eddies, and X is the forcing from the small-scale eddies, like gravity wave drag.  G is the total zonal drag force.  F = jFy + kFz , where the two components are given by:

 

We can best understand how this all works by starting with a simple model and progressing to more complex models. 

 

How big is this forcing?  We can get an idea from our knowledge of the magnitude of fov*.  fo ~ 10-4 s-1.  Air moves from the tropics to the poles, about ¼ of Earth’s circumference, in ~2 years.  Thus, in steady-state, G ~ 10-4 s-1 x .2 m s-1 = 2x10-5 m s-2.

 

First assume no seasonal cycle.  In this case, all time derivatives = 0, and the equations of motion become:

 

 

­

 

The Coriolis force due to the residual meridional velocity just balances the eddies due to large and small eddies.  The residual adiabatic cooling is just balanced by diabatic heating.  If the eddy forcing didn’t exist, then the residual meridional velocity would be 0 and the meridional drift would stop.

 

The two relationships are connected by the continuity equation, which gives the relationship:

 

­

If the eddy forcing does not exist, then the diabatic heating and the residual vertical velocity must also be 0.  The simplest model of the atmosphere is thus one that is in radiative equilibrium, with the zonal mean temperature being equal to the steady-state radiative balance.

 

Next, we assume a diabatic heating that mimics the annual solar cycle.  In this model, we parameterize the diabatic heating as the departure of the stratospheric temperatures from the radiative equilibrium value:

 

where αr is the Newtonian cooling rate.

 

The stratosphere’s temperature response will lag the temperature forcing because of the air’s thermal capacity.  This lag is small compared to the annual cycle because the thermal relaxation time is 5-20 days.  Otherwise, this model produces an annually varying temperature distribution that looks very much mike radiative equilibrium.

If we substitute this annually varying diabatic heating rate into the continuity equation, we find the equation:

 

where we assume that the density change is the most important change with altitude.

 

where tr is the inverse of ar. 

 

So, we see that the eddies drive the stratosphere away from thermal equilibrium and the radiation tries to drive it back.

 

The largest departures from radiative equilibrium occur in the:

  • polar winter stratosphere
  • summer mesosphere
  • winter mesosphere.

 

How does all of this determine the mean meridional circulation?  In the wintertime, mesosphere, internal gravity waves propagate from the troposphere into the mesosphere, where they break.  These breaking waves deposit energy, resulting in a strong zonal force that acts like friction, slowing down the zonal velocity.  Stationary planetary waves play a similar role in the wintertime stratosphere.  

 

Consider the equation:

In the Northern Hemisphere winter, fo > 0 and G = wave drag is a westward force and thus <0.  Thus the residual meridional velocity must be >0.  By mass continuity, the residual vertical velocity must be negative, and air descends and warms.

 

In the Southern Hemisphere summer, fo < 0 (the zonal wind is westward, easterlies).  As a result, G > 0 (eastward force to opposing the wind)

Thus –(-|fo|)v* = G implies that v* > 0 , or northward drift.

 

Linear Waves

 

Wave classifications

 

  1. restoring mechanism
    1. buoyancy – gravity waves
    2. rotation (Coriolis force) – inertial waves
    3. mean meridional potential vorticity gradient – Rossby waves (or planetary waves)
  2. sources
    1. forced waves – heating / orography
    2. free waves – normal modes of atmospheric oscillations
  3. vertical / meridional structure – all waves propogate zonally
    1. propagating – in y or z directions
    2. trapped (evanescent) – in y or z directions

 

Introduction to wave theory.

 

Assume a shallow fluid wave. 

(diagram)

 

 

 

 

h(x,t) = H + h’(x,t)

momentum equation:

continuity equation:

We assume a harmonic solution, as always occurs for linear wave theory:

where

 

To stay on the wave’s crest, which gives us an idea of the wave’s phase speed, we choose f so that this happens.  We substitute these wave equations into the momentum and continuity equations and get the dispersion relations:

 

 

To satisfy these two simultaneously, we must have:

The phase speed, c, is given by:

This relationship is non-dospersive, since c does not depend on k.

 

Buoyancy waves (Gravity waves).  These are vertically propagating waves.  Assume first that the mean zonal wind is zero.

(diagram)

 

 

 

 

 

 

In this case,

If you displace the parcel along a slantwise path, it will stay along that path because the gravity force is down, but the pressure force acts to move it back along the path.

The vertical force, component of the vertical buoyancy force along the path are:

Thus, the resulting description of the motion is the simple harmonic oscillator:

which has the solution:

where k is the wave number in the zonal direction and m is the wavenumber in the vertical direction.

where Lx ~ 10-100 km, Lz ~ 5-15 km, and the period is ~ minutes to an hour.

 

The phase speed in this case is dependent on both k and m, making it dispersive, meaning that the wave packet changes shapes as it proceeds:

The group velocities, which tell us about energy propagation, is found by taking the derivation of the frequency w with respect to k and m:

When the phase is being propagated downward and to the east, the group velocity and hence the energy is propagated upward and to the east.  We can see this by taking certain signs of the solutions:

cx > 0 leads to cgx > 0, but cz < 0 leads to cgz > 0

The group velocity is perpendicular to the phase velocity.

 

What is the source of gravity waves?

 

The atmosphere can be excited at frequencies only up to the Brunt-Vasaila frequency, N.  If the frequency is less than N, then we will get waves that are excited on the slant paths.

 

Assume now that there is a mean zonal wind.

Now the group velocities are:

The group velocity is increased to the east by the addition of the mean zonal wind.

 

It should be pointed out that the actual air molecules are not moving very far from the position that they would have with the mean zonal wind.  In the stratosphere, this motion is no more than a 100 meters or so.

 

Inertio-gravity waves.  Inertio-gravity waves are gravity waves that have wavelengths long enough (~100’s of km) that they are significantly affected by the Coriolis force.  In the Northern Hemisphere, the Coriolis force deviates the waves so that they turn clockwise with height.

(diagram)

 

 

 

 

 

 

 

 

 

 

In the momentum and continuity equations, we must introduce terms for the Coriolis force.  When we do this, the dispersion relation that we achieve is:

where l is the wavenumber in the meridional (y) direction.

 

For the waves to exist, |f| < |w| << N.

 

Inertio gravity waves tend to propagate more horizontally than do internal gravity waves.

 

 

Planetary waves (Rossby waves). 

  • These waves are most important for stratospheric transport.
  • While there can be either stationary planetary waves that are forced by orography and heating or traveling free waves, forced by ???, the stationary planetary waves are the most important.
  • The wave forcing is the isentropic gradient of potential vorticity (i.e., the change of f with latitude).  Steady planetary waves conserve potential vorticity, just as steady buoyancy waves conserve potential temperature.
  • Consider the conservation of vorticity: h = z + f   (eta = zeta + f).  Assume that zinitial = 0.  Assume that the air parcel moves to another latitude by dy.  By conservation of vorticity, znew + fnew = finitial, which implies that

where b = df/dy = the planetary vorticity gradient.

For dy < 0, the rotation is cyclonic (counterclockwise), znew > 0, because