Download 57 4. ATMOSPHERIC RESPONSE TO OCEAN

Variabilité climatique dans les Tropiques. Cours de Claude Frankignoul 2012-2013
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4. ATMOSPHERIC RESPONSE TO OCEAN FORCING
4.1 The tropical atmosphere
The atmosphere is compressible and has a relatively large coefficient of thermal
expansion, meaning that if T increases at constant pressure,
decreases. The air is a mixture
of gases and the perfect gas law
p = ρ RT
(4.1)
applies to all its components. For instance R = 287 J kg-1 K-1 is the gas constant for dry air.
The air is moist, mostly in the lowest few kilometers, but the water vapor content (typically
0.5 %) is so small that the pressure is given to a good approximation by that of dry air. Water
vapor is lighter than air, so moist air is lighter than dry air, so that the gas constant for moist
air is larger than for dry air. Rather than use a gas constant that is a function of the water
content, one can use a virtual temperature in (1) – the temperature that dry air must have in
order to have the same density as the moist air at the same pressure. It turns out that the
virtual temperature is at most (in warm, moist regions) larger than the temperature by only a
few degrees (see Wallace and Hobbs, 2006; Sarachick and Cane 2010).
At equilibrium, the rate of evaporation is equal to the rate of condensation, and the air
is “saturated” with water vapor. The partial pressure of water vapor e is then the saturated
vapor pressure es, and it only depends on T. To a good approximation, es increases
exponentially with temperature,
, where A = 6.11 h Pa and
= 0.067 °C-1 are
constant and T in in °C, a simplified statement of the Clausius-Clapeyron relationship (see
Marshall and Plumb 2008). As a consequence, the moisture content of the atmosphere
decreases rapidly with height, and it is larger in the tropics than at high latitudes.
There are other important differences with the ocean. The atmosphere is more strongly
stratified (typically N ~ 8’ and R1 ~ 3000 km), and there is no free surface at its top, which
strongly influences vertical propagation. The atmosphere has clouds and is primarily heated
from below. The incoming solar energy is mostly absorbed by the ocean and the land, while
the radiation to space mainly occurs in the upper troposphere. As a result, the surface tends to
be warmer than the atmosphere above, which is unstable to convective motions.
To understand atmospheric convection, the compressibility of the atmosphere must be
taken into account. Indeed, consider an adiabatic displacement of an air parcel in a dry
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atmosphere. If the parcel rises, its environment has a lower pressure. To adjust, the parcel will
expand, doing work on its surrounding, which cools the parcel. Hence the temperature varies
during adiabatic displacements. This can be quantified by applying the first law of
thermodynamics to a parcel of air of unit mass with a volume V. If a differential increment of
heat dq is exchanged by the parcel with its surroundings by thermal conduction or radiation,
one has
(4.2)
where Cv dT is the change in internal energy due to the temperature change dT,
is the
external work done by the parcel in expanding by a volume dV, and Cv is the specific heat at
constant volume. Since we are dealing with a parcel of unit mass,
the specific volume. Using the relation
, where
is
, and relation (1), (2) can be written
(4.3)
For adiabatic motions (no condensation) , dq = 0, so that (3) yields
(4.4)
Since the atmosphere is in hydrostatic equilibrium, on has
, where dz is a small
displacement. Replacing in (4) yields
(4.5)
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≈ 9.8 K km-1 is called the dry adiabatic lapse rate. It is the rate of change of a
where
parcel of dry air that is raised adiabatically.
The potential temperature
of an air parcel is the temperature that the parcel would
have if it were compressed or expanded adiabatically from its existing pressure and
temperature to a standard pressure p0 (generally taken as 1000 mb). Replacing
in (4) by (1)
yields
(4.6)
Integrating (6) from p0 where
to p gives
Cp T
p
ln = ln
p0
R θ
or the Poisson’s equation
θ = T(
p0 R/C p
)
p
(4.7)
Relations (5) can be used to determine whether a parcel being displaced adiabatically upward
!"
in an environment with actual lapse rate 𝛤 = −( )! will experience a restoring force and
!"
tend to return to its initial level, or will continue to raise. The temperature of the environment
at
z + δz
is given for small
! dT $
by T + # & δ z = T − Γδ z
" dz %E
and the density by
ρ E (z + δ z) = p(z + δ z) / RT (z + δ z) . On the other hand, the parcel at
also has pressure
p(z + δ z) but has temperature Tp = T − Γ dδ z and density given by ρ p (z + δ z) = p(z + δ z)/ RTp .
Hence, the parcel will experience a restoring force if ρ p > ρ E (z + δ z) or
neutral if
and unstable if
; it will be
. A dry atmosphere will thus be unstable if
temperature decreases faster with height than the adiabatic lapse rate. Correspondingly, it is
easy to show that the dry atmosphere will be unstable if its potential temperature decreases
with height. This should not persist in the free atmosphere, since the instability would be
immediately eliminated by strong vertical mixing (convection). It could only persist near the
ground in the presence of strong heating from below. The lower tropospheric lapse rate is
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≈ 4.6 K km-1. Thus there should be no instability nor convection: the tropical atmosphere is
always stable to dry convection, as sketched in Fig. 4.10 from Marshall and Plumb (2008).
Because of the moisture content of air, however, convective instability may frequently
occur. We can express the moisture content of air by introducing the specific humidity q,
given by the ratio of the mass of water vapor to the mass of air per unit volume,
where
,
is the total mass of air. In the absence of mixing, evaporation, and
condensation, q is conserved by a moving air parcel since the mass of both water vapor and
dry air must be separately conserved.
The saturation specific humidity q* is the specific humidity at which saturation occurs.
Since both water vapor and dry air behave as perfect gases, one has
(4.8)
where
is the saturated partial pressure of water vapor at temperature T (see Fig. 1.5
above) and Rv the gas constant for water vapor. At fixed p, q* strongly increases with T.
The relative humidity RH is the ratio (in percentage) of the specific humidity to the
saturation specific humidity
. In the lower tropical atmosphere, RH is close
to 80%, which is rather close to saturation. Hence, a parcel of moist air may not need to be
raised very much before it reaches its condensation level. Indeed, when the air parcel is lifted,
the pressure decreases and the parcel cools. Decreasing the pressure should increase q*, but
decreasing T has a stronger effect in view of the exponential T-dependence of es, so that q*
decreases rapidly with altitude. Since q is conserved, the condensation level
where q = q*
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is rapidly reached, and above it excess water condenses so that q remains equal to q*. As q*
decreases with height, q will decrease correspondingly, which is visible as convective clouds.
The condensation releases latent heat, which partly compensates the cooling due to adiabatic
expansion. Hence, above zc the temperature of the parcel falls off more slowly, until neutral
buoyancy is reached at zt, the cloud top (Fig. 4.18 from Marshall and Plumb 2008). In the
tropics, the air is warm and moist, so that the cloud top is high, typically reaching near the
850 hPa level. Note that below the condensation level, the temperature of a rising parcel of air
decreases with height at the dry adiabatic lapse rate until it become saturated with water
vapor. Above the condensation level, temperature is given by the saturated adiabatic lapse rate
discussed e.g. in Wallace and Hobbs (2006) and Marshall and Plumb (2008).
Most convection in the atmosphere is moist convection with saturation and cloud
formation. As downwelling air parcels do not become saturated and descend slowly, moist
convection takes place in narrow, cloudy updrafts. They are surrounded with much larger
areas of clear, slowly descending dry air cooled by radiative cooling, even in large-scale
convective cloud systems. The more common convection in the tropics is rather shallow and
topped by cumulus clouds (the trade cumulus inversion). The towers are typically 1-2 km high
and grow very rapidly in about 15’, transporting heat upward. Deep penetrative convection is
found in large cumulus convection cells or hot towers which may reach about 175 hPa, just
below the tropopause, where it is so cold that the cloud top is sheared off by the wind and
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forms the anvil canopy made of ice cristal (Fig. 4.20 of Marshall and Plumb 2008). In the
tropics, there are typically a few thousand hot towers, but even in active disturbances
convection only covers a small fraction of the area, as illustrated for a convective system in
the ITCZ. They provide the main mechanism for transporting heat vertically from the surface
to the free troposphere (see Marshall and Plumb 2008 for more details). The deep clouds
produce heavy rain and are frequent in the west Pacific. On the other hand, convective clouds
tend to reach a lower height in the east-central Pacific ITCZ or the tropical Atlantic, and they
are associated with lower rain rates.
The observations show that organized deep convection in the tropics mainly occurs
where the sea surface temperature is above 27°C, even though it is not a sufficient condition
since it also requires a strong wind and low-level moisture convergence to sustain the
convective activity. This condition is often met in the western tropical Pacific and in the ITCZ
(Gutzler and Wood, JCLI 1990), resulting in a large monthly rain rate. This is illustrated in
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Fig. 1 of Black and Bretherthon (JCLI 2009), who show the distribution of surface
convergence, vertical velocity near the trade cumulus inversion, and precipitation. There is a
good correspondence with the SST distribution, but convergence and, to a lower extent,
rainfall occur in a narrower band than would be expected from the SST distribution. As
discussed below, convergence is more closely related to sea surface temperature gradients.
The analysis of Lin et al. (JCLI 2006) of 6 months of satellite observations between
30°N and 30°S illustrates more precisely the link between the tropical deep convective
systems (DCS) and SST on a daily basis. Note that if the deep convective systems are most
frequent where the sea surface temperature is about 302 K (Fig. 1), they are most frequent per
unit area at higher temperature (Fig. 6). Particularly striking is the link between sea surface
temperature T and surface wind or moisture convergence in Fig. 13 and 15 (convergence
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mostly occurs for T ≥ 28°C), which stresses that local evaporation does not suffice to generate
a strong convective activity.
4.2 Equatorial waves
To investigate wave propagation in the tropical atmosphere, we consider for simplicity
an isothermal dry atmosphere and assume hydrostatic equilibrium, which is valid at large
scale. We consider a small perturbation in a basic state at rest. The density of the basic state
depends on height, since (1) is written
(4.9)
p0 = ρ 0 RT0
and the hydrostatic approximation
€
yields
(4.10)
Variabilité climatique dans les Tropiques. Cours de Claude Frankignoul 2012-2013
where
65
is the scale height, typically 7 or 8 km. It is useful to write the
thermodynamic energy equation in term of p rather than T. To do so, we replace dT in (3) by
, which gives
or, introducing
,
(4.11)
The linear, forced equations for the perturbation are
(4.12)
(4.13)
(4.14)
(4.15)
(4.16)
where
is the heating rate or diabatic heating. The boundary conditions are
and a radiation condition (outgoing energy flux) for
at
. It is convenient to transform the
variables by
(4.17)
This leads to the system of linear equations
(4.18)
(4.19)
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(4.20)
(4.21)
(4.22)
Eliminating
between (20) and (22), and then between (20) and 21) yields
(4.23)
(∂ z −
1
g
)p‘t = −
w‘+ g(u‘x + v‘y + w‘z )
2H
2H
(4.24)
where w’ can be eliminated to give
(4.25)
Free solutions
If
, the system (18), (19) and (23) admits separable solutions of the form
Replacing in (23) yields
(4.24)
so that one must have
(4.25)
and
(4.26)
where h is the equivalent depth. The vertical eigenfunctions are defined by
(4.27)
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with, from (23)
at
(4.28)
or outgoing radiation condition when
Note that the boundary condition at z = 0 is easier to establish in log-pressure coordinate (Wu
et al 2000). The system (27) – (28) defines a complete set of eigenfunctions Rn in the vertical,
as (2.19) – (2.22) in the oceanic case. However, as shown by Lindzen (1967), Holton (1975),
and Wu et al. (2000), in the atmosphere there is no discrete vertical mode because of (28),
except for the Lamb wave (sometimes called external mode) that travels with the speed of
sound and has a large equivalent depth
, zero vertical displacement (the velocity is
parallel to the surface), and a pressure perturbation that decays exponentially with height
(4.29)
Rossby waves with this structure propagate toward high latitudes when excited in the tropics
and play a role in the ENSO teleconnections.
Except for the Lamb wave, the system (27) – (28) only has for solution a continuous
ensemble of eigenfunctions (also called the internal or baroclinic modes) that either propagate
vertically or will be trapped in the vertical, and have the form
(4.30)
where n is any positive real number. Assuming a time dependence of the form
that the continuous modes of wavelength
shows
propagate vertically as the atmosphere is
semi-infinite. They cannot survive since their energy is constantly radiated into space. Hence,
in a forced problem, the continuous modes rapidly disappear and only the barotropic mode
and the forced modes remain. Note that this distinction between free and forced modes is not
seen in a model atmosphere with a lid.
The horizontal propagation of the modes is governed by (26) and
As in the oceanic case, we look for solutions if the form
(4.31)
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This lead to
(4.32)
which only has bounded solutions (the parabolic cylinder functions) for
(4.33)
with m = 0, 1, 2, … This relation can be solved for h to determine the equivalent depth hm as a
function of k,
, and m, which are determined by the initial conditions or the forcing. This
differs from the oceanic case where the vertical eigenfunctions determine the equivalent
depth, which then controls the horizontal propagation. It is linked to the continuous
distribution of the vertical wavelength of the baroclinic modes due to the semi-infinite
atmosphere. From (33), one gets the equivalent depth or the gravity wave speed (ghm )1/2
1/2
" 4(k 2 + k β / ω )ω 2 %
(2m +1)β
(ghm ) = − 2
(1± $1+
'
k + kβ / ω
β 2 (2m +1)2 &
#
1/2
(4.34)
where only the – solution is valid on a sphere (the other solution does not decay fast enough at
high latitude). At low frequencies
, (34) simplifies into
,
(4.32)
which is independent on k.
Forced solution
The forced response to diabatic heating is obtained by expanding the forcing into
parabolic cylinder function as in (31), namely setting
(4.33)
where J(z) is the vertical distribution of the diabatic heating, and
determines k,
and thus hm. The forcing then appears in the equation for the vertical structure of mode m
1
1− γ
d 2 Rm
γ −1
+(
−
)Rm =
J m (z)
2
2
γH
dz
γ Hhm 4H
(4.34)
,
Variabilité climatique dans les Tropiques. Cours de Claude Frankignoul 2012-2013
Let
. If A is real, thus if
propagates vertically. If A is imaginary, thus if
69
is small and positive, the forced mode
is either negative or very large, the
response is vertically trapped with a decay scale A.
At low frequencies
,
solution thus propagates vertically, with
is small and positive as seen from (32). The forced
for a downward phase velocity and
upward group velocity. One has
(4.35)
The vertical group velocity is given by
(4.36)
The vertical propagation is very slow. For a 30-day period and m=0, the largest wave, the
vertical group velocity is about 2 m/day and the vertical wavelength 150 m. Lower
frequencies propagate even more slowly. Note that at high frequencies, the vertical
propagation is faster, and vertically propagating Kelvin and mixed Rossby-gravity waves
have been observed in the equatorial stratosphere and play a role in the Quasi-Biennial
Oscillation.
This shows that the vertical propagation is very slow at the low frequencies
characteristic of SST forcing (less than 60 m / month). As a result, the response is confined to
the heating region and the dominant vertical scale is that of the heating. Of course, dissipation
of momentum and Newtonian cooling must be taken into account to treat the problem
correctly. Wu et al. (2000) have shown that the relative importance of the two types of
damping strongly affects the zonal propagation and the structure of the response.
4.3 Atmospheric response to sea surface temperature anomalies
One can distinguish two aspects of the problem, even though they are intimately
coupled. The first one is to establish the atmospheric response to prescribed tropical diabatic
heating anomalies, which will determine both the response of the large-scale tropical
circulation and the teleconnections to the mid-latitudes. The second aspect is to establish the
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diabatic heating anomalies caused by given sea surface temperature anomalies. To understand
the ocean – atmosphere modes of variability such as seen in the ENSO phenomenon, the
emphasis should be on the surface wind stress response since wind stress primarily controls
the sea surface temperature changes and thus the air-sea coupling.
Over the tropical oceans, low-level moisture convergence is the dominant source of
moisture for precipitation, and it generally exceeds local evaporation. Hence, deep convection
should occur where the surface winds are convergent. However, latent heating in deep
convective system causes upward motion, which must be associated with wind and moisture
convergence near the surface, so that the two aspects are coupled. To distinguish between
cause and effect, one can compare the relative importance of the atmospheric boundary layer
processes versus the deep-tropospheric processes in determining the surface wind and the
moisture convergence, given the sea surface temperature distribution.
Although the cloud base is above the trade cumulus inversion, the diabatic heating due
to precipitation in deep convective systems seems to have a rather continuous profile, as
shown in Fig. 9 from Hagos et al. (JCLI 2010). Using soundings, satellite observations and
microphysics, as well as atmospheric reanalysis, they showed that the temporal variability of
the heating profiles is dominated by two modes: a top-heavy profile (mode 1) and a bottomheavy profile (mode 2). In the deep mode, which is more frequent in the west tropical Pacific,
the diabatic heating is maximum at about 400 hPa and is small in the boundary layer. On the
other hand, the diabatic heating peaks at about 700 hPa in the shallow mode, which is more
frequent in the east Pacific and there is a substantial heating in the atmospheric boundary
layer. Note that the net diabatic heating includes smaller changes in the radiation fluxes, and
sensible heating in the atmospheric boundary layer.
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The observations show that the surface wind anomalies during ENSO events are
largely decoupled from the wind anomalies above the trade cumulus inversion, say at 850 hPa
and above. As first suggested by Lindzen and Nigam (1987), low-level convergence is
primarily controlled on seasonal and longer time scale by the influence of sea surface
temperature gradients on boundary layer temperature and surface pressure gradients, rather
than by the free-tropospheric component and downward momentum mixing. More refined
analysis (e.g. Black and Bretherton, JCLI 2009) shows that the climatological surface winds
and the boundary layer convergence are indeed primarily driven by the sea surface
temperature gradients, even though the climatological surface zonal surface winds are
predominantly associated with deep diabatic heating and downward momentum mixing. At
higher levels, the circulation is primarily driven by the deep convective systems.
To a reasonable approximation, the surface wind convergence due to deep tropospheric
heating and that induced by sea surface temperature gradients can be studied separately.
However, the two mechanisms are not independent since it is the anomalous low-level
moisture convergence that feeds the deep convection and the release of latent heat during
precipitation.
Gill’s model
Gill (1980) introduced a very simple model of the steady response of the tropical
atmosphere to deep diabatic heating that has been widely used. The main assumptions are
- the response can be explained by the linear equations around a basic state at rest with
mechanical damping and radiative cooling.
- The heating is taken to be a half sinusoid, which is obtained by considering a normal
mode. The vertical distribution of the response is assumed to be that of the diabatic
heating.
- The vertical distribution of the response is assumed to be that of the diabatic heating.
This implicitly assumes that there is a rigid lid at the troposphere.
The governing equations are
fu + p y + ε v = 0
(4.38)
Variabilité climatique dans les Tropiques. Cours de Claude Frankignoul 2012-2013
with
. Gill uses c = 70 m/s. The vertical velocity is simply given by
72
. In the
model, the Rayleigh damping and the Newtonian cooling are taken to be identical and, to get
realistic zonal scales, very strong,
≈ 1.5 days (for realistic friction the zonal scales would
be much too large).
The response to equatorial forcing with a gaussian meridional distribution that projects
onto the first Rossby mode and the Kelvin mode, and is centered around x = 0 and has the
form cos kx for
and zero otherwise, is illustrated in Fig. 1 (Gill, 1980). Here the
signature of a forced damped Kelvin wave is seen to the right (eastward group velocity) and
that of a damped symmetric Rossby mode (westward group velocity) is seen to the left. The
Kelvin mode shows equatorial easterlies extending eastward a distance
mode shows equatorial westerlies extending westward a distance
and the Rossby
, as expected in a
steady state and consistent with the faster propagation speed of the Kelvin wave. Although the
solutions are too zonal and the strong eastward extension is not observed, nor simulated in
more sophisticated models, the model broadly reproduces many observed features, or features
simulated by atmospheric general circulation models (GCMs) provided a very strong damping
is chosen, ε =1.5 days. Because of its analytic simplicity, the model has very often been used.
−1
Nonetheless, there are a number of problems. The heating, which primarily takes place above
the cloud base, is artificially brought down to the surface. The surface wind is thus similar to
the wind in the lower troposphere, which is not observed. The model only has one vertical
mode, but the implicit rigid lid approximation is arbitrary, and its height influences the
solution. Note that the model has been generalized to many vertical modes, which somewhat
reduces the sensitivity to the lid height. Finally, the damping times are much too short for
tropospheric motions.
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Experiences with another simple linear model that has different Rayleigh damping (2
d-1) and Newtonian cooling (30 d-1), which is more realistic, show that in practice the response
to an El Niño anomaly depends on the diabatic heating anomaly in the whole tropical band,
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not solely where the sea surface temperature anomaly is maximum (Neelin, QJRMS 1988). In
these calculations, the diabatic heating is prescribed. However, it is the SST that should be
prescribed, and the diabatic heating calculated.
Application of the Gill’s model to SST forcing
Gill’s model has often been used in many theoretical studies. Zebiak (MWR 1986)
have used it as atmospheric component of the coupled model of Cane and Zebiak (Science
1985), which has been successfully used to make ENSO simulations and forecasts. To do so,
Zebiak implemented a nonlinear scheme that simulates the dominance of low-level moisture
convergence in sustaining deep convection. Gill’s model is used in an anomaly mode with
prescribed climatology of temperature and wind. The diabatic heating anomaly induced by an
SST anomaly is estimated iteratively by adding to the diabatic heating due to evaporation the
heating due to moisture convergence. The latter is estimated as a function of the wind
climatology and the wind anomaly driven by the diabatic heating. The procedure is repeated
iteratively until a steady state is reached, and the surface wind stress is then used to drive the
ocean model that predicts the SST. A given SST anomaly will thus have a different influence
in regions of mean subsidence or mean upward motion. This representation of the CISK
mechanism turns out to be efficient in amplifying the initial heating anomaly, so that the
model gave rather realistic results, as illustrated below for the response to the mature phase of
a typical El Nino event: (a) composite SST anomaly (b) composite wind anomaly (c) surface
wind response of the model without the iterative feedback or (d) with it (e) observed surface
wind divergence, and that modeled without (f) and with (g) feedback. Note that the strong
easterlies to the east of the maximum heating, a common weakness of linear atmospheric
models, are not observed (composites in Fig. 3 below of Okumara et al. JCLI 2011).
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More refined models show that the vertical structure of the diabatic heating is
fundamental to the structure of the forced atmospheric response. This is discussed by Wu et
al. JAS 2000), who used the vertical eigenfunction analysis for a semi-infinite atmosphere
given in section 4.1. The solution critically depends on relative importance of momentum
damping and thermal damping, and it also depends on the length of the domain when the
damping is not strong (Wu et al. JAS 2001). For weak damping, the solution needs to be
considered in a zonally cyclic domain since the response then reaches a very large domain.
This is illustrated below. Figure 2 shows (left) the diabatic heating profile that they used, plus
in dashed line how it enters (4.33), and (right) how the heating projects onto the (continuous)
vertical eigenfunctions. Note that they are all excited, but with variable intensity. Figure 3
shows the response for strong friction, strong cooling and slow gravity wave speed, Fig. 4, the
response for the same damping but faster gravity wave speed, resulting in much larger
spreading. Finally, Fig. 5 shows a case with weak friction and weak cooling. Further
sensibility studies show that significant surface winds can only be driven by an elevated heat
source when thermal damping is more important than momentum damping. Otherwise, the
response remains confined to the heating region. In addition, the model is not able to
reproduce the strong moisture convergence needed to sustain the precipitation linked to the
diabatic heating profile. Hence, other mechanisms must be responsible for the moisture
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convergence. Figure 1 p. 63 shows that the distribution of precipitation and moisture
convergence is narrower that that of SST. Hence, this suggests that it is not SST that plays the
dominant role, but SST gradient, as demonstrated by Lindzen and Nigam (1987).
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Boundary layer response to SST gradients
Lindzen and Nigam (JAS 1987) suggested that on large scales the low-level
convergence was predominantly determined by the imprint of the SST gradients on the
atmospheric boundary layer, hence SST gradients were the main cause of tropical deep
convection. Their main assumption is that SST anomalies produce sensible and evaporative
heat fluxes, which generate (virtual) temperature anomalies in the atmospheric boundary
layer. The latter is assumed well mixed as it is heated from below. However, Lindzen and
Nigam included the stratocumulus-cumulus cloud layer in their definition of the mixed layer,
which extended up to 700 mb and was capped by a strong temperature inversion at a height of
about 3 km. The model does not apply to the zonally averaged (Hadley) circulation, which is
known to be strongly influence by vertical momentum mixing between the free troposphere
and the boundary layer. As discussed in Black and Bretherthon (JCLI 2009), the use of a very
deep boundary layer implies a SST influence that goes to high, but it was needed to get
realistic sea level pressure gradients (as they neglected the sea level pressure gradient driven
by the free troposphere).
Lindzen and Nigam noticed that the “eddy” temperature field (deviation from the
zonal mean) was generally well correlated below 700 hPa, hence they assumed the form
𝑇 𝑥, 𝑦, 𝑧 = 𝑇 𝑦, 𝑧 + 𝑇!! (𝑥, 𝑦)(1 − 𝑧
!
!!
)
(4.39)
where temperatures are virtual temperatures (temperature that dry air would have if its
pressure and density were those of the moist air), γ is an order-one constant, Hb is the
reference height for the boundary layer (3 km), and one has
𝑇 𝑦, 𝑧 = 𝑇 𝑦, 0 − Γ𝑧
(4.40)
with Γ the undisturbed lapse rate. At sea level, 𝑇!! is given by the SST. The top of the
boundary layer (taken to be the 700 hPa level) is at
𝑧! = 𝐻! + ℎ! + ℎ
(4.41)
where ℎ! is the small zonal mean deviation from 𝐻! and ℎ the small remaining eddy
contribution.
In this model, the SST drives the virtual temperature and leads to a pressure gradient
that set the surface circulation. Integrating the momentum equations over the boundary layer
yields for the mass-weighted boundary layer transport perturbation
Variabilité climatique dans les Tropiques. Cours de Claude Frankignoul 2012-2013
80
−𝑓𝑉! + 𝜀! 𝑈! = −𝜙!
(4.42)
𝑓𝑈! + 𝜀! 𝑉! = −𝜙!
(4.43)
with 𝜀! !! of the order of 1 or 2 days, which is realistic, and
𝑈! =
!!
!
𝜌𝑢𝑑𝑧
!! !(!! ) !
𝜙! =
!!
!
𝑝! 𝑑𝑧
!! !(!! ) !
≈!
!!
!
𝜌𝑢𝑑𝑧
!
!(!
)
!
!
(4.44)
(4.45)
From hydrostatic equilibrium, one finds the averaged pressure
𝜙 ≈ 𝑔 1 + Γ
!
!!
!,!
ℎ − 𝐴 𝑇!! (𝑥, 𝑦)
(4.46)
where A is a constant. The first contribution is the pressure perturbation due to the change in
boundary layer thickness, and the second is that due to the density change caused by the SST.
A main assumption is that in regions of deep convection, any surface mass
convergence is rapidly vented by convection, so that the full effect of the first term in (46) is
not realized. This “back pressure” effect needed to get realistic values is incorporated by
assuming that the boundary layer relaxes back to equilibrium on a short time scale εT-­‐1, taken to be about 30’, which corresponds to the development time of cumulus convection −𝐻! 𝑈!" + 𝑉!" ≈ 𝑤 𝑍! = 𝜀! ℎ (4.47) This is unrealistically short and not applicable in region of subsidence. Nonetheless, the
model generally gives realistic surface winds. It may explain how SST initiates deep
convection by a mass flux at the top of the boundary layer, but a CISK model is needed to
simulate the free tropospheric response. Interestingly, replacing h in (47) by (46) yields an
equation very similar to that of the Gill’s model, although the physics are different.
In summary, in the Lindzen-Nigam model, the atmospheric boundary layer is
assumed to be mixed, so that the temperature throughout is correlated to the SST. Warm
boundary layer temperature hydrostatically causes low pressure, and the pressure gradients
lead to frictional convergence. The PBL top rises causing a back pressure to inhibit the
convergence, and damping is accomplished by venting the rising boundary layer to the free
troposphere. Although the model has flaws (to get realistic winds, the BL height needs to be
too large, and the venting (damping) is too large), more sophisticated models confirm that
SST gradient play a much larger role than deep convection in generating the surface
convergence. This is shown by the comparison in Black and Bretherthon (Fig.5, JCLI 2009)
Variabilité climatique dans les Tropiques. Cours de Claude Frankignoul 2012-2013
81
between the surface winds and the convergence driven by the SST gradient alone (top) and
that which including the pressure gradient from the free atmosphere, derived from the NCEP
reanalysis at 850 hPa. Clearly, the contribution of the latter is small in the central and eastern
Pacific and in the tropical Atlantic, where the two figures are nearly undistinguishable.
A particularly successful application of the Lindzen-Nigam model is to the WES
feedback (wind – evaporation – SST feedback), which plays a key role in tropical dynamics.
Suppose there is a warm SST anomaly north of the equator. There will be cross-equatorial
wind toward the warm SST. However, away from the equator, the Coriolis force deflects the
wind toward the right in the northern hemisphere. As a result, the northeasterlies are
weakened, which reduces the evaporation and enhances the SST. In the southern hemisphere,
the opposite occurs as the southeasterlies are strengthened. A climatological example is given
in Fig. 3 below. Note that the wind field favors upwelling along South America but prevents it
to the north, consistent with the northward position of the ITCZ.
Sensitivity studies with atmospheric GCMs show that the tropical atmosphere
responds nearly deterministically to SST forcing, as illustrated in Fig. 3 from Fennessy et al.,
(MWR 1985) based on an earlier comparison of a (coarse) GCM to the SST anomaly in
January 1983 (Fig. 1) : the initial conditions affect little the model response.
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AGCM with prescribed SST (AMIP runs) show that the surface wind field is wellreproduced at low frequencies, suggesting that the surface winds indeed respond primarily
Variabilité climatique dans les Tropiques. Cours de Claude Frankignoul 2012-2013
83
deterministically to the SST distribution. An early example with a T21 AGCM forced by the
“observed” SST between 1970 and 1985 is reproduced below, from Latif et al. (JCLI 1980).