Goal: Use understanding of physically-relevant scales to reduce...
Transcript of Goal: Use understanding of physically-relevant scales to reduce...
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Goal: Use understanding of physically-relevant scales to reduce the complexity of the
governing equations
• Scale analysis relevant to the tropics [large-scale synoptic systems]*
*Reminder: Midlatitude scale analysis in dynamics
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Momentum Equations in Spherical Coordinates
!
dudt"uv tan#Re
+uwRe
= "1$%p%x
+ 2&v sin# " 2&wcos# + Frx
dvdt
+u2 tan#Re
+vwRe
= "1$%p%y
" 2&usin# + Fry
dwdt
"u2 + v 2
Re
= "1$%p%z
+ 2&ucos# " g + Frz
total derivative pressure gradient
Coriolis gravity friction
To what extent are these terms important?
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Scale Analysis [following Holton §11.2]
• Goal: To determine relative importance of the terms in the basic equations for particular scales of motion.
• Approach: Estimate the following quantities 1) The magnitude of the field variables. 2) The amplitudes of fluctuations in the field
variables. [To estimate derivatives.] 3) The characteristic length, depth and time scales
on which these fluctuations occur.
But first a coordinate change…
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Vertical coordinate transformation It is often convenient to replace height z by pressure p. Consider p=p(x,z), where p(x,z) is assumed to be monotonic in z and x represents any horizontal coordinate, and the system of level curves sketched below:
x
z Along each level curve, p(x,z) = constant. So let’s consider the differential of p:
dp = 0 = ! p!x z
dx + ! p!z x
dz! !z!x p
= "! p /!x
z
! p /!zx
p1
p2
p3
Under hydrostatic equilibrium: !z!x p
=! p /!x
z
"g
For further convenience, we introduce the geopotential:
!
" = g dzz0
z
#
Thus: !h" p= !#1!h p z
The notation ∇h denotes horizontal components of the gradient operator.
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Conservation of mass in pressure coordinate
Consider a parcel of mass δm which is conserved following the motion of the flow:
!
1"m
d("m)dt
= 0
Assuming a density ρ, δm = ρδV, and: 1
!"Vd(!"V )dt
= 0 = g("x"y"p)
ddt
"x"y"pg
!
"#
$
%&
!
! 0 = 1!x
d!xdt
+1!y
d!ydt
+1!p
d!pdt
"
#$
%
&'
(
)*
+
,-=
!u!x
+!v!y
+!"!p
"
#$
%
&'
!!u!x
+!v!y
+!"! p
"
#$
%
&'= 0
No density, hence no explicit time-dependence appearing in the mass conservation relationship expressed in pressure in the vertical.
! !dpdt
Pressure velocity
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“log pressure” coordinates For the purposes of scale analysis, we’ll consider the natural logarithm of pressure rather than pressure itself:
z*= !H ln pps
"
#$
%
&'
where ps is a standard reference pressure and H is a scale height defined as:
!
H "RdTsg
In this coordinate: w*= dz*
dt= !H d
dtln p
ps
"
#$
%
&'= !
H!p
!
ddt"##t
+ vh $ % + w * ##z*
!"! p
=!w*!z*
!w*H
For an isothermal atmosphere at temperature Ts [~global surface temperature], z*=z.
However, for realistic temperature profiles in the troposphere, the difference between z* and z is usually small.
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Scaling quantities
Scale Symbol Value Horizontal
velocity U 10 m s-1
Horizontal length L 106 m
Vertical depth H 104 m
Time τ~L/U 105 s
Vertical velocity W TBD Geopotential
fluctuation δΦ TBD
Temperature fluctuation δT TBD
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Physical Parameters
!
g "10 ms#2
Re "107m
$0 = ?f = 2%sin$& =10#5 m2 s#1
gravity
radius of earth
viscosity
We’ll consider both middle and tropical latitudes….
Coriolis parameter
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Continuity + Horizontal Equations
!
dudt"uv tan#Re
+uw *Re
= "$%$x
+ 2&v sin# " 2&w *cos# + Frx
dvdt
+u2 tan#Re
+vw *Re
= "$%$y
" 2&usin# + Fry
!
"u"x
+"v"y
+"w *"z*
#w *H
= 0
!
WH !
"u"x
+"v"y
#UL
In the horizontal, for nondivergent flow, ∇h⋅v=0.
!
W "UHL
To perform the horizontal scaling, we’ll consider the magnitudes of terms relative to the horizontal advective scale:
!
U 2
L
!
1
!
LRe
~ 0.1
!
LWReU
"HRe
~ 10#3
!
"#U 2
!
fLU
= Ro"1 !LU
= Re!1 ~10!12
Ro is the Rossby Number
Re is the Reynolds Number
!
LW"cos#U 2 $
"HU
~ 0.1
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Continuity + Horizontal Equations
!
dudt"uv tan#Re
+uw *Re
= "$%$x
+ 2&v sin# " 2&w *cos# + Frx
dvdt
+u2 tan#Re
+vw *Re
= "$%$y
" 2&usin# + Fry
!
1
!
LRe
~ 0.1
!
LWReU
"HRe
~ 10#3
!
"#U 2
!
fLU
= Ro"1 !LU
= Re!1 ~10!12
For mid-latitudes (λ~45°), f~10-4, so Ro ~ 0.1. Thus, the Coriolis term (~10) must be balanced by the geopotential gradient term, implying:
!
"# ~ fUL [~1000 m2s-2] At low [tropical] latitudes, f≤10-5, so Ro ≥1, so Coriolis may not balance the pressure gradient force. In fact, for Ro ≥10, we expect:
!
"# ~U 2 [~100 m2s-2]
Thus, for synoptic disturbances in the tropics, geopoential perturbations are an order of magnitude smaller than for similar-sized systems in mid latitudes, which has several important consequences…
!
LW"cos#U 2 $
"HU
~ 0.1
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Vertical equation
!
dw *dt
"u2 + v 2
Re
=gRdTH
#$#z*%
& '
(
) * "1
+ 2+ucos, " g + Frz
!1!" p"z
= !1!""" p#
$%
&
'(
!1"""z
= !g!""" p#
$%
&
'(
!1 !!! p
=!!!z*
!z*! p
= "Hp!!!z*
!"1!" p"z
=gHp!
"#"z*$
%&
'
()"1
=gRdTH
"#"z*$
%&
'
()"1
!
WUL
"U 2HL2
~ 10#4
!
U 2
Re
~ 10"5
!
U"cos# $10%3
!
10
!
"WH 2 #
"UHL
~ 10$14
!
gRdTH
"#"z*$
% &
'
( ) *1
The above scaling implies hydrostatic equilibrium:
!
"#"z*
=RdTH
But it also provides a scaling for horizontal temperature perturbations*: !T = H
Rd
""z*
!! ~ !!Rd~ U
2
Rd~ 0.3 K
It is assumed that the geopotential consists of a mean component, which is a function of the vertical coordinate only, and a synoptic perturbation; since we’re evaluating synoptic scale motion, the scaling applies to the perturbation part.
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(Dry) Thermodynamic equation
!
""t
+vh#$h
%
& '
(
) * T +
w *HN 2
Rd
=JcP
For δT~0.3 K, the first term on the left-hand side is of order 0.3K x 10-5 s-1 ~ 0.3K day-1. When precipitation is absent, the diabatic heating term on the right-hand side comprises long-wave radiative emission, which is observed to be of order 1 K day-1. Thus,
For the tropical tropopshere, N-1~100 s, so: !
w *HN 2
Rd
"Jcp
# w* " Jcp
Rd
HN 2
!
w* ~W ~ 0.3 cms"1
In the absence of precipitation, synoptic scale tropical vertical velocities are much smaller than in similar-sized extratropical systems. Also, from the continuity equation, the divergence of horizontal wind is of order 10-7; thus the flow is essentially nondivergent.
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(Brief overview of) precipitating tropical synoptic systems
What modifications are anticipated for precipitating tropical synoptic systems? For precipitating tropical synoptic systems, rain rates of 20 mm day-1 are typical. For a unit area cross-section, this precipitation rate implies a daily condensation of 20 kg of H20. [Think density x area, which is dimensionally mass/length.] Since Lc~2.5 x 106 J kg-1, the net condensational energy input per day into a unit area atmospheric column is:
ddt
mH2OLc( ) ! 5"107 Jm#2day#1
Assuming this heating is spread uniformly over the entire tropospheric depth, or over a unit area column of tropospheric air mass, implies a daily heating rate per unit air mass, i.e., , of: (ps ! pt )g
!1
Qcnv =Jcnvcp
=ddt mH2O
Lc( )gcp(ps ! pt )
" 5 Kday!1
As we’ll see later, the distribution of condensational heating via tropical convection is nonuniformly distributed in the vertical and tends to maximize in the mid- to upper-troposphere (~300 to 400 mb), with heating rates of order 10 K day-1. Recalling that the radiative heating rate is ~ 1 K day-1, the above implies a vertical velocity scale ~ 10x larger than in nonprecipitating regions. This in turn implies a relatively large component of the divergent flow.