Models with reduced vertical structure

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Models with reduced vertical structure

• Adam Sobel
• Banff Summer School

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Vertical discretizations, 0 N-level system
Level models are derived by numerical
discretization of the equations, by a
finite-difference method or something close to
it a mathematical, rather than a physical
argument. We can then just choose to use a small
number of levels.
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Vertical discretizations, 1 N-layer system
Consider a stack of N layers of fluid, each of
constant density ?i, and assumed to move with
velocity ui, both uniform within each layer. Let
the top surface be either free, or rigid. Mean
layer thicknesses Hi, displacements ?i
?N
?N, uN
The motions of the system can be expressed in
terms of the ui, and ?i. We get a set of
shallow-water equations, coupled through the
interface displacements (which affect the
pressure gradients)

H
?3, u3
?2, u2
?1
?1, u1
H1
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Vertical discretizations, 2a Modal
decomposition from the layers
Alternatively, we can decompose the motion into
modes. We label the modes with indeces. The
gravest or barotropic mode has all ui equal,
and has the largest surface displacement (or
surface pressure if rigid lid). The next, or
first baroclinic has the lower half moving in
sync, out of phase with the upper half. The
higher order modes have increasingly finer
vertical structure. Let the amplitude of
each velocity mode be Uj(x,t).
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Vertical discretizations, 2b Modal
decomposition from the layers
The modes are complete we can reconstruct the
layer motions with ui?j uij Uj(x,t). The uij
are the vertical structures they tell us how
much layer i moves per unit amplitude of mode j.
Under linear dynamics, the modes evolve
independently. Each one satisfies its own set of
shallow water equations with an appropriate
equivalent depth Hi, which generally decreases as
vertical structure becomes finer. Linearly ?t
Uj fk Uj -gr?j ?t ?j Hj r Uj 0 The
gravity wave phase speed of the mode is
cj(gHj)1/2
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Vertical discretizations, 3 Modal decomposition
from a continuous fluid
The modal decomposition can also be applied to a
fluid with continuous basic stratification ?0
(z), as long as we still have a well-defined
upper boundary (free or rigid). Then the modes
are continuous functions, an orthogonal and
complete basis set, forming a discrete spectrum,
e.g., sines cosines for N2 -(g/?0)d?0/dzconst.
Again we can reconstruct the local
motions, u(x,z,t)?uj(z)Uj(x,t) where the sum now
goes (countably) to infinity, and again the
modes satisfy independent equations if the
dynamics are linear.
U2
U1
U0
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Vertical discretizations, 4a Application (or
lack thereof) to the atmosphere
This is all well and good, for the
oceanographers. Without a well-defined upper
boundary, it doesnt work. Physically, modes are
standing waves which arise through reflection
off the boundaries. In general, we have
vertically propagating modes. They dont become
standing if they have nothing to reflect them.
Strictly speaking, the atmosphere has no
vertical modes. If you want to do some kind of
decomposition, you can decompose into
horizontal eigenfunctions of the linear equations
on the sphere (Laplaces equation) and then solve
a vertical structure equation. This is tidal
theory, and is fine, but doesnt lead to
models with reduced vertical structure.
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Vertical discretizations, 4b Application (or
lack thereof) to the atmosphere

• However, it isnt that bad in all cases. Depends
  on frequency. High-frequency waves have high
  vertical group velocities and get into the
  stratosphere. Low-frequency modes have small
  group velocities and are damped quite effectively
  before getting too far. Thus they tend to have
  large amplitude only near the forcing, and are
  effectively vertically trapped. Assuming a rigid
  lid for steady, thermally forced circulations is
  thus actually not so bad.

?z cgz ?, where ? damping timescale
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Vertical discretizations, 4c Application (or
lack thereof) to the atmosphere
The problem doesnt apply only to modal
decompositions. Say we want to work directly with
layers, say, make a model for a single layer of
the atmosphere, defined as that bounded by two
isentropic surfaces, ?1, ?2. We still have the
problem that those surfaces will be perturbed by
motions outside the layer, and we cant formulate
an appropriate boundary condition without either
a) solving the equations of motion outside the
layer (in which case whats the point) or b)
making an ad hoc assumption. In practice, we do
the latter, in the spirit of hierarchical
modeling we isolate the dynamics that can be
captured with a single layer, and then think
later about how it interacts with the rest of the
atmosphere.
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Single layer models, 1 - general

• Isentropic layer, ? 1
• Heating that is uniform in the layer changes ?1
  and ?2 equally, so thickness of layer doesnt
  change
• Heating with nonzero vertical gradient changes
  layer thickness
• Height or pressure of layer, whose gradient
  appears in momentum equation, can be perturbed by
  motions above and below

dz/dt(d?/dt)(d?/dz)-1
?2
?1
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Single layer models, 2- barotropic model

• Simplest model there is
• Assume heating has no vertical gradient, or there
  is none at all any forcing must be mechanical
  (though possibly induced by thermal effects in
  other layers)
• Pure vorticity dynamics
• (?t ur) (? f) forcing dissipation
• r2u ?
• No layer thickness/thermodynamic equation
• only free modes are Rossby waves

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Barotropic model applications, a) the first
weather forecast model!
It didnt work too badly, for very short times.
Once you get past a day or two, baroclinic
effects (interaction of different layers) become
important.
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Barotropic model applications, b) dynamics of a
barotropic polar vortex
Absolute vorticity ? f
Juckes McIntyre 1987, Nature 328, 590-596 It is
an initial value, or spin-down problem here
nothing to maintain the vortex against
dissipation.
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Looks like obs
potential vorticity ¼ (? f)??/?p
McIntyre Palmer 1983, Nature, 305, 593-600
This is Rossby wave breaking nonlinear
saturation of the Rossby wave, the only linear
mode supported by the barotropic vorticity
equation
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Single layer models, 3 shallow water model for
a single layer

• Equivalent to allowing free surface now 3 time
  derivatives (u, v, h) and 3 wave modes (1 Rossby
  2 gravity)
• Now we can allow the heating to have a vertical
  gradient. Thus the bounding surfaces ?1 and ?2
  change height at different rates, effectively
  creating a mass source for the layer this is
  thermal forcing, appearing in equation for layer
  thickness
• ?t h r(uh) Q
• Vorticity equation now has a vortex stretching
  term, because divergence ? 0
• ?t ur(? f) (ru) (? f) forcing
  dissipation
• or, (? t ur)(? f)/hforcing
  dissipation
• (? f)/h is the potential vorticity for the
  system

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A shallow water polar vortex calculation
material lines
PV map
Thermal forcing allows vortex to be
maintained against dissipation.
Polvani, Waugh Plumb 1995, JAS 58, 1288-1309.
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Rossby wave breaking is really a 3D process
Polvani and Saravanan 2000, JAS 57, 3663-3685
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Single layer models, 4 shallow water model for
a single mode

• First baroclinic mode only
• Assume rigid lid at tropopause - not so bad for
  steady motions, very bad for high frequencies
  (they propagate fast vertically)
• Now heating itself (not vertical gradient)
  appears effectively as a mass source
• Linear dynamics typically assumed not strictly
  necessary, but formally consistent with
  assumption of single mode
• Very popular model for large-scale tropical flows
  (Matsuno 1966, Webster 1972, Gill 1980)
• Forms the atmosphere part of intermediate ENSO
  models e.g. Cane-Zebiak (with simple heating
  parameterization) surface wind is the important
  output in this case

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Single layer models, 4 shallow water model for
a single mode

• First baroclinic mode is arguably justified from
• observations, for tropical tropospheric
  circulations
• (T) tend to have deep, single-signed vertical
  structure
• u, v tend to change sign once, with maxima near
• bottom and top of troposphere. This is a
  consequence
• of the structure of the heating (not a true
  mode).

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Single layer models, 4 shallow water model for
a single mode
low-level geopotential wind, equatorially
centered heating
Rayleigh drag
off-equatorial heating
Newtonian cooling (p / T here)
Imposed heating, fn of lat long
Gill 1980, QJRMS 106, 447-462
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Some of many possible criticisms

• Rayleigh drag on first baroclinic mode is
  inappropriate parameterization of surface drag,
  as it acts equally on upper lower tropospheric
  winds (need to add barotropic mode to cure this),
  and too strongly at that for typical parameters
• Often used as a model for surface winds, but some
  argue that free-tropospheric heating shouldnt
  produce surface winds, because vertical group
  velocity is too small, thus damping will kill the
  waves before they get to surface though there
  are counter-arguments to this
• Heating must be given somehow. This is the
  hardest part of the problem (convective
  parameterization)

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Another model for surface winds Lindzen and
Nigam (1987), JAS, 44, 2418-2436.

• Atmospheric boundary layer, with no pressure
  gradient at its top extreme alternative to
  assumption that free-tropospheric heating drives
  surface flow
• Assume that within ABL, TSST
• Hydrostatically, then rp r(SST), also ABL
  depth, h
• Assume strong drag (reasonable, since near sfc)
• Linear momentum equation gives flow (if SST
  known)
• Any mass convergence/divergence must be balanced
  by ascent/descent across ABL top
• Equations are formally equivalent to Gill model,
  but different interpretation of parameters
  (Neelin 1989)

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What controls precipitation?
Some Gill-type models use quasi-equilibrium
convective schemes PCAPESST. Higher SST makes
near-sfc air have higher T, q, thus greater
convective instability, more rain. This is
thermodynamic control of deep convection.
Depends on local SST (compared to some
large-scale average)
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What controls precipitation?
In Lindzen-Nigam model, by contrast, rainfall is
implicit in ABL mass convergence. That implies
mass export to free troposphere. Assuming fixed
ABL relative humidity, moisture export is implied
as well. In steady state, reasonable to assume
that moisture is rained out. That implies
rainfall over SST maxima. rp -rSST,
and ignoring rotation ?u rp. Since P -ru, we
have P-r2 (SST). Curvature, rather than local
value, of SST is whats important in this
model. So we have two arguments for rainfall
over SST maxima, but one relies on absolute value
of SST, the other on local curvature. Totally
different physics.
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Two-layer models, 1 Baroclinic instability

• Can get two vertical degrees of freedom any way
  you want modes, layers, levels.
• If you have sufficient baroclinicity, rotation,
  and stratification, you will get baroclinic
  instability.
• Arguably the simplest model for this important
  phenomenon
• See Kushners lecture for more on this

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3-Layer models

• Main application of which I am aware is hurricane
  modeling see excellent review by Arakawa J.
  Climate,17, 24932525 (covers convective
  parameterization issues generally)
• In such models, need baroclinic mode to represent
  convectively-driven circulation as in
  climate-type models, but now ABL physics is
  absolutely essential.

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Two-layer models, 2 Quasi-equilibrium tropical
circulation model (Neelin and Zeng, 2000,JAS, 57,
1741-1766)

• Two modes, barotropic and 1st baroclinic
• Baroclinic mode not simple sinusoid derived by
  assuming T(p) has moist adiabatic structure
  observed in tropics. This implies vertical
  structure of geopotential also, thus u - result
  still looks qualitatively like Gill mode
• Now, surface drag can be applied appropriately
  surface wind is reduced by cancellation of
  barotropic and baroclinic modes, rather than just
  damping of the latter
• Simple set of parameterizations for moist
  physics, based on GCM schemes, but projected on
  2-mode system
• consistent moisture and energy budgets

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Comparison to obs seasonal cycle of precip in
QTCM