Lesson 6
Shallow
Where are we going…
and Hydrostatic Approximations
– What are they?
– What are they used to obtain?
Primitive
Vector form of EOM
System of Equations
Spherical form of EOM
– Be able to identify the terms
Cartesian
(Rectangular) Coordinates
– Be able to write and identify the terms of the
EOM in this coordinate system
f-plane
and β-plane
Shallow
Cartesian (rectangular)
Primitive
Equations
f-plane
β-plane
– What are these approximations?
Shallow (Traditional)
Approximation
Used
to scale our EOM
result when combined with the
hydrostatic approximation is our set of
Primitive Equations
The
– The basis of our ocean and atmospheric
numerical models
Recall in Spherical Coordinates
Zonal
(East-West) distance
dx = a cosφdλ
Meridional
(North-South) distance
dy = adφ
Vertical
distance
dz = da
1
Shallow (Traditional)
Approximation
Complete
Now scale the remaining terms in
the vertical equation
u2
a
v2
a
1 ∂P
ρ ∂z
g
2Ωu cos φ
spherical transformation of EOM
Du uv tan φ uw
1
∂P
−
+
=−
+ 2Ω(v sin φ − w cos φ ) + ν∇ 2u
Dt
a
a
ρ a cos φ ∂ x
Dv u 2 tan φ vw
1 ∂P
+
+
=−
− 2Ωu sin φ +ν∇ 2v
ρa ∂ y
Dt
a
a
Dw u 2 v 2
1 ∂P
− − =−
− g + 2Ωu cos φ +ν∇ 2 w
ρ ∂a
Dt a a
Shallow (Traditional)
Approximation
Complete
spherical transformation of EOM
102
6378000
102
~
6378000
~
~ 10
~ 10
~ (10−4 ) (10 )(1) = 10−3
Shallow (Traditional)
Approximation
Resulting Shallow Approximation Equations
Du uv tan φ uw
1
∂P
−
+
=−
+ 2Ω(v sin φ − w cos φ ) + Fλ
Dt
a
a
ρacos φ ∂λ
Du
u
1
∂P
− (2Ω +
)v sin φ +
= ν∇ 2u
Dt
a cos φ
ρ a cos φ ∂ x
Dv u 2 tan φ vw
1 ∂P
+
+
=−
− 2Ωusin φ + Fφ
a
ρa ∂φ
Dt
a
Dv
u
1 ∂P
+ (2Ω +
)u sin φ +
= ν∇ 2 v
Dt
a cos φ
ρa ∂ y
Dw u 2 v 2
1 ∂P
− − =−
− g + 2Ωucos φ + Fa
Dt
a a
ρ ∂a
∂P
= −ρ g
∂z
2
Hydrostatic Approximation
Primitive System of Equations
When
the vertical scale of the disturbance
is much less than the horizontal scale, we
can neglect the vertical acceleration
Without friction, the dw/dt term becomes
the hydrostatic equation
0=
1 ∂P
+g
ρ ∂z
=0
Cartesian (Rectangular)
Coordinates
Need to neglect the change in position vectors
from our EOM…
Dkˆ u
v
So we get:
1 ∂P
ρ ∂z
– An EOS ( p = ρ RT for atmosphere )
– Continuity Equation ( ∇ ⋅ u = 0 for incompressible fluid )
– Thermodynamic Energy Equation
Cartesian (Rectangular)
Coordinates
Dv
u
1 ∂P
+ ( f + tan φ )u +
= ν∇ 2 u
Dt
a
ρ a ∂y
PLUS:
note: now we can no longer
predict vertical motion
Diˆ
u
=
(sin φˆj − cos φkˆ )
Dt acos φ
of both approximations
Du
u
1
∂P
− ( f + tan φ )v +
= ν∇ 2 u
Dt
a
ρ a cos φ ∂x
g+
Important
Result
Dt
=
a
iˆ +
a
ˆj
Dˆj
utan φ ˆ v ˆ
=−
i− k
Dt
a
a
DU Du ˆ Dv ˆ Dw ˆ
=
i+
j+
k
Dt
Dt
Dt
Dt
In our EOM the curvature terms on the left hand
side can be dropped…
The
resulting Cartesian EOM:
Du
1 ∂P
=−
+ fv − 2Ωw cos φ +ν∇ 2u
Dt
ρ ∂x
Dv
1 ∂P
=−
− fu + ν∇ 2v
Dt
ρ ∂y
Dw
1 ∂P
=−
− g + 2Ωu cos φ + ν∇ 2 w
Dt
ρ ∂z
3
β-plane
f-plane
For disturbances with small horizontal scales
(~100 km), the tangent plane we assume in our
Cartesian system is the f-plane
Due to small North-South disturbances we say:
f = 2Ω sin φ = f 0
– Where f0 is a constant based on the center of the
area
– i.e. we assume constant φ for the region
For large areas with varying φ (over tens of
degrees) between the mid-lats and
Equator we call the tangent plane in our
Cartesian system a β-plane
D
Dφ 2Ωcos φ
• Where: β = Dy (2Ωsin φ ) = 2Ωcos φ Dy = a
•
• β: variation of Coriolis parameter with latitude
•
• f 0 : f at middle latitude of region
• β y = ∂f : value at the middle latitude of region
This can be used for large regions near the
poles as well because f is near its maximum
and varies slowly
∂y
Lesson 6 - Part 2
Scale
Analysis
Scale Analysis
Simplifies the EOM by neglecting the
unimportant terms in the governing equations for
particular types of motion
This filters out some unwanted types of solution
– What is it and how is it used?
Scaling
horizontal EOM
– Be able to obtain a basic balance of terms
Geostrophy
– Be able to identify the basic balance for large
scale motions in the atmosphere and ocean
– Ageostrophic wind
Variation of f is: f = f 0 + β y
– Shows us which terms are the most important
– The complete EOM describes all types and scales
of fluid motion (i.e. sound waves, gravity waves,
Rossby waves, etc)
– We filter out the scale(s) we do not want as it may
obscure the scale(s) we do want
4
Scale Analysis
Typical
values of the following are
specified:
– Magnitudes of the field variables
– Amplitude of fluctuations of these variables
– Characteristic length, depth and time
scales on which the fluxes occur
Scale Analysis: Example 1
Estimating
– Typical mid-latitude synoptic cyclone
– Surface pressure may change (fluctuate)
10 hPa over a horizontal distance of 1000
km
• Pressure flux: δP
• Horizontal length scale (x or y): L
– Magnitude of the horizontal pressure
gradient:
∂P
∂P δP 10 hPa
and
≈
=
= 10−2 hPa m−1
∂x
∂y
L 10 3 m
Scale Analysis: Method
Focus
on a particular motion (equation)
Find the characteristic values (magnitude)
Plug into the equation to find the order of
magnitude for each term
Determine the terms to be neglected
Result: simpler equation
Scale Analysis: Example 2
Estimating
– Incompressible continuity equation
– Want to estimate the vertical motion in the
open ocean…
Variable
Scale
Estimate
|u|~|v|
U
0.1 m/s
∆x~∆y
L
107 m
∆z
H
103 m
|w|
W
?
5
Scale Analysis: Horizontal EOM
Scale Analysis: Horizontal EOM
Atmosphere
Atmosphere
Define
characteristic scales of field
variables based on observed values
U ~ 10 m/s
W~ 1 cm/s
L ~ 106 m
D = H ~ 104 m
(δP/ρ) ~ 103 m2 s-2
T ~ L/U ~ 105 s
g ~ 10 m/s2
f ~ 10-4 s-1
Horiz velocity
Vert velocity
Length
Depth/height
pressure change (h)
Advective time scale
Gravity
Coriolis parameter
ν = 10-5 m2/s
a ~ 107 m
For air
Earth’s radius
2
3
4
5
6
−2Ωv sin φ −2Ωw cos φ
uv tan φ 1 ∂P
−
−
ρ ∂x
a
ν∇ 2 u
dv
dt
−2Ωu sinφ
uv
a
u 2 tan φ
1 ∂P
−
a
ρ ∂y
ν∇ 2 v
U2
L
f0U
x equation
y equation
Scales
Magnitudes 10-4
10-3
f 0W
10-6
UW
a
U2
a
δP
ρL
νU
10-8
10-5
10-3
10-16
Scale Analysis Horizontal EOM:
Atmosphere
Atmosphere
Result
Coriolis
PGF
1 ∂P
2uΩsin φ = −
ρ ∂y
Note: No time! Therefore a diagnostic
expression
– i.e. cannot use to predict the evolution of a velocity
field
7
uw
a
Scale Analysis Horizontal EOM:
– The dominate balance is between 2 and 6
1
du
dt
This
H2
balance is Geostrophic Balance
– Diagnostic expression
– Fundamental balance for mid-latitude
synoptic scale flow
– Component Expressions:
− fv g = −
1 ∂P
ρ ∂x
fug = −
1 ∂P
ρ ∂y
6
Scale Analysis: Horizontal EOM
Ocean
The result is still geostrophic balance;
however the order of magnitude is different!
– 10-5 vs. 10-3
U ~ 1 cm/s
W~ 10-5 m/s
L ~ 107 m
D = H ~ 103 m
f ~ 10-4 s-1
ν = 10-6 m2/s
Horiz velocity
Vert velocity
Length
Depth/height
Coriolis
parameter
For ocean
Geostrophic Wind
The
zonal geostrophic wind depends on
the meridional gradient of pressure
The meridional geostrophic wind
depends on the zonal gradient of
pressure
In vector form:
1 ˆ
k × ∇P
Vg ≡ ug iˆ + v g ˆj =
ρf
Departure From Balance
Ageostrophic Wind
Quasi-geostrophic
The
equations
– Prognostic equations
• Retain the time derivative
du
1 ∂P
− fv = −
dt
ρ ∂x
dv
1 ∂P
+ fu = −
dt
ρ ∂y
portion of the wind that departs
from geostrophy
i.e., the departure from geostrophic
balance
Vag = V − Vg
dV
= − fkˆ × Vag
dt
kˆ dV
Vag = ×
f
dt
7
Lesson 6 - Part 3
Dynamic Similarity
Dynamic
Flows
Similarity
parameters
Non-dimensional
– Reynolds Number
– Rossby Number
– Ekman Number
of different scales or different
properties can be found to be
dynamically similar if their relevant
parameters have the same magnitude
Thus, conclusions about one flow can
be extended to dynamically similar flows
Non-dimensional Parameters
Scaling
Allow
Consider
an efficient measure of the
relative importance of various terms in
the dynamical equations
Useful in approximations and
simplification
the horizontal EOM
– Scale
each term
dVH
1
= − ∇ H P − 2Ω× VH + ν∇ 2HVH
dt
ρ
U2
T
P
ρ0L
f 0U
υU
L2
Let’s
ignore PGF and calculate the ratio
of different terms to each other...
8
Reynolds Number (Re)
Determines dynamic stability
Ratio of terms:
– Acceleration (or inertial terms)
– Friction
Rossby Number (Ro)
Re =
UL
ν
Estimate for relative strengths of inertial and CF
Ratio of terms:
– Acceleration (or inertial terms)
– Coriolis force
Re ~ 1 - both terms are important
Re << 1 (~10-2) - friction dominates
Ro is small - Coriolis dominates
– Can therefore neglect the inertial term
– Laminar flow
Ro is large - Inertial dominates
Large values indicate turbulent flow
Examples
– What is Re in the open ocean?
– What is Re for the atmosphere (synoptic scale)?
Ro =
U
f 0L
– Flow is nearly geostrophic
– Planetary effects can be ignored
Examples
– What is Ro in the open ocean?
– What is Ro for the atmosphere (synoptic scale)?
Ekman Number (Ek)
Describes
the layer in which friction
balances coriolis
ν
Ek =
Ratio of terms:
f 0 L2
– Friction
– Coriolis Force
Small
Ek disturbances can propagate
Ek disturbances dissipate due to
friction
Examples
Large
– What is Ek in the open ocean?
9