Space Science: Atmospheres
Part- 8b
Read 9.3 H
Viscosity
Eddy Mixing
Planetary Boundary Layer
Ekman Spiral
Spin Down Time
Reminder
r r
r r
Corliolis Force = - 2 " #xv = - 2 "# (kxv)
Steady State (Geostrophic $ large scale motion)
r r
Fa = Force Applied = 2 "# (kxv)
In Northern Hemisphere
r r
Can write : zˆ % Fa = 2 "# zˆ % (kxv) = - "f v
where zˆ is the local normal to the plane of flow
or
v = - [zˆ % Fa ]/ "f = [Fa % zˆ ]/ "f
v is to the right of applied force (f > 0)
opposite in the southern hemisphere
Near surface: we need to add in
drag forces
Result:
Winds near surface are often to
the left of the winds aloft
This defines the ‘boundary layer’
Planetary Boundary Layer
Surface is not smooth on
some scale
Scale of ‘roughness’ differs
over mountains, plains, cities
Typical average ~1km (a fraction of H)
treat turbulence induced as a viscosity
Ekman number not negligible
E-1 = Re = L V / ν ; ν = kinematic viscosity
Viscosity
transport of turbulence (momentum)
between adjacent layers of fluid
Molecular--primarily above homopause
Eddy --below
Using viscosity (like diffusion) means:
at some length scale we do not
describe \the details of the motion
Planetary Boundary Layer (cont.)
Ekman number not negligible
E-1 = Re = L V / ν ; ν= kinematic viscosity
Viscosity (transport of turbulence)
Write the velocities as an average + a fluctuation
u = u + u'
v = v + v'
w = w + w', w' - fluctuations
Average velocities : u , v,w
Therefore, averages of fluctuations
u ' = v' = w ' = 0
Substitute u, v, w into momentum + continuity equations
and then average : Get equations for u , v, w
plus
cross terms uu' etc = 0
plus
"
terms like
( # u'w')
"x
that are not zero
Viscosity(cont.)
Meaning of " u'w' (this is a momentum flux)
Coupling of vertical and horizontal velocity
fluctutations (turbulence)
Such terms give, for example, the vertical transport of
horizontal motion
Often write it as
%u
%z
That is - -if the horizontal flow (u) is the same at all z
- - then there is no effect - -as much random motion
(momentum) moves up as down : this defines $
*
" u'w' # - " $
Dimensions : $ ~ Lv
L is a characteristic mixing length and v is a mean speed
or v = L/t , estimate a mean mixing time instead
Near the planetary surface : use roughness scale and speed
Eddy viscosity
$ # K & 1 - 100 m2 /s
$ & 10 -5 m2 /s
applies primarily in the thermosphere
Molecular viscosity
Viscosity(cont.)
" u'w' ( a momentum flux)
Terms in momentum equation are
$
the negative gradient of a flux # ( " u'w' ) This is then - $z
the change in momentum per unit volume per unit time
$
$u
[" &
]
$z
$z
$2
or
% "& 2 u
$z
Some effort to show
r
r
dv
Viscous term
"
% "&' 2 v
dt
%
use 'K' now and drop the bar from u
1D
1 dp
$u/$t = 0 = + K d 2u/dz 2
" dx
assume u(z = 0) = 0 at the surface
dp
=0
dx
u(z) = u o (z o - z)
u(z o ) = u o and
No Coriolis Force (Equatorial)
u
Very Near Surface
z
ρK[∂ u/∂ z] ~ const
Momentum Flux due to turbulence/ mixing
" = - # u'w' = #K(du/dz)
" is a stress (units of pressure)
Reach a steady state with Constant Stress between layers
but length/ mixing scale changes near the surface
K~Lv
L $ %z
near surface; % von Karman const.
' &u *
v$ L) ,
( &z +
' &u * 2
- # u'w' = # L ) ,
( &z +
2
2
'
*
&u
Solution : # (% z) 2 ) , = " ; the shear stress
( &z +
then
u = ( u * / % ) ln(z/z o )
z o = surface roughness
1/2
u * = ("/# )
~ u'w'
1/ 2
Ekman Spiral
Sec. 9.3H
Include Coriolis,
viscosity and pressure gradients
(3 – D)
"u
1 "p
"2
= 0 = fv #
+K 2u
"t
$ "x
"z
"v
1 "p
"2
= 0 = # fu #
+K 2 v
"t
$ "y
"z
2 – D Flow, but coupled vertically
!
Boundary conditions
u(0) = 0 = v(0)
u (z $ # ) = u g , v(z $ # ) = v g
Geostrophic values from
1 !p
f vg =
% !x
1 !p
f ug = "
% !y
Use geostrophic solutions
"2
K 2 u = #f(v # v g )
"z
"2
K 2 v = f(u # u g )
"z
Trick : Define
V $ u + iv
Multiply v equation by i and
add the two equations
"2
K 2 (u + i v) = i f [ (u - u g ) + i(v # v g )]
"z
"2
K 2 (u + i v) = i f [ u + iv - (u g + iv g )]
"z
OR
"2
V = % (V # V g )
2
"z
% $ if/K
Solution (cont)
V "V g = a e
1)
2)
#1/2z
+ be
"# 1 / 2 z
V(0) = 0
V ($) = V g
# 1/2
3)
% i f (1/ 2
1+ i
1/ 2
= ' * = + (1+ i) : [i =
]
&K)
2
% f (1/ 2
+ = ' *
& 2K )
V = V g + a e+(1+ i)z + b e+(1+i)z
,
Use
(2)
V = V g + b e+(1+ i)z
Use
(1)
0 = Vg - b
b = Vg
,
V = V g [ 1 - e -+z e i+z ]
Analytic Solution!
decay and periodic terms
Solution (cont)
Need to go back to u, v
For simplcity : u g = u g
(V = u + iv)
vg = 0
Solutions
u = u g [ 1 - e -"z cos "z]
v = u g e -"z sin "z
Ekman was actually an oceanographer
Example " = [ f /2K]1/ 2
# = 30 o Latitude
then
f = 7 $ 10 -5 /s
If
K = 100 m2 /s
" -1 = 1.6 km
" -1 = Thickness of Planetary Boundary Layer
Ekman Spiral
Solutions
at z >> γ-1
u=ug, v=0
u/ug
v/vg
Ekman Spiral
Wind speed and direction
vs. altitude
v (m/s)
z in meters
u (m/s)
Dashed: Measurements
Solid: Analyitc solution
mid latitude
Force Balance
If at 10 km, Northern hemisphere
1000 mb
1005 mb
p
ug
Cor.
1010 mb
Below ~ 1 km
1000 mb
v
v
p
1005 mb
1010 mb
ug
fric
cor
!
Surface winds in the Northern Hemisphere
tend to blow to the left of winds aloft.
Ocean Layer Driven by
Wind Shear (Ekman)
Ekman ‘Pumping’
The coupling of the geostrophic
layer (often circular motion) to
the surface through the planetary
boundary layer gives a spin down
time for the circular motion in the
troposphere
Define a measure of circular motion : vorticity, "
r
r
r
" = k • # $ v (# $ v = 2%zˆ , 2 - D cir. motion)
Use Ekman solution and continuity to calculate
&"
' (" / ) s ;
&t
) s is a time constant for dissipating
the geostrophic flow by coupling to the surface
Not too hard to show (see H p132)
τs ~ 2γH/f = (2/fK)1/2 H
Mid latitudes ~ days
Summary of terms
•
•
•
•
•
Viscosity: definition
Eddy Mixing
Planetary Boundary Layer
Ekman Spiral
Spin Down Time