3 Spatial Averaging
3-D equations of motion
Scaling => simplifications
Spatial averaging
Shear dispersion
Magnitudes/time scales of
diffusion/dispersion
Examples
Navier-Stokes Eqns
Conservative form momentum eqn; x-component only
⎛ ∂ 2u ∂ 2u ∂ 2u ⎞
1 ∂p
∂u ∂ 2
∂
∂
+ (u ) + (uv) + (uw) − fv = −
+ υ ⎜⎜ 2 + 2 + 2 ⎟⎟
ρ ∂x ⎝ ∂x ∂y ∂z ⎠
∂t ∂x
∂y
∂z
1
2
3
4
5
1 “ storage” or local acceleration
2 advective acceleration
3 Coriolis acceleration [f = Coriolis param = 2ωsin(θ), θ = latitude]
4 pressure gradient
5 viscous stress
Turbulent Reynolds Eqns
u = u + u ' , etc (u = time average)
Insert & time average (over bar)
∂
∂ 2 ∂ 2
[(u + u ')(u + u ')] = u + u '
∂x
∂x
∂x
e.g., x-component; term 2
∂u ∂v ∂w
+
+
Continuity
∂x ∂y ∂z
∂u ∂ 2
∂
∂
+ (u ) + (u v ) + (u w ) − fv
∂t ∂x
∂y
∂z
1
2a
x-momentum
3
⎞ ∂ ⎛ ∂u
∂ ⎛ ∂u
1 ∂p ∂ ⎛ ∂u
⎞
2⎞
=−
+ ⎜υ
− u ' ⎟ + ⎜⎜υ
− u ' v' ⎟⎟ + ⎜υ
− u ' w' ⎟
ρ ∂x ∂x ⎝ ∂x
⎠ ∂y ⎝ ∂y
⎠
⎠ ∂z ⎝ ∂z
4
5
2b
Specific terms
Term 2a: could subtract u times continuity eqn
⎡ ∂u ∂v ∂w ⎤
+
=u⎢ +
⎥
∂
∂
∂
x
y
z
⎣
⎦
∂u
∂u
∂u
+v
+w
u
∂x
∂x
∂x
Terms 5 + 2b
NC form of momentum eqn (term 2)
x-comp of turbulent shear stress
∂u
∂u
2
2
− u ' = τ xx ≅ −u ' ≅ 2ε xx
υ
∂x
∂x
⎛ ∂u ∂v ⎞
∂u
− u ' v' = τ xy ≅ −u ' v' ≅ ε xy ⎜⎜
+ ⎟⎟
υ
∂y
⎝ ∂y ∂x ⎠
∂u
⎛ ∂u ∂w ⎞
− u ' w' = τ xz ≅ −u ' w' ≅ ε xz ⎜
+
υ
⎟
∂z
⎝ ∂z ∂x ⎠
εxx, εxy, εxz are
turbulent (eddy)
kinematic viscosities
resulting from
closure model (like
Exx, Exy, Exz)
Specific terms, cont’d
Similar eqns for y and z except z has gravity.
For nearly horizontal flow (w ~ 0), z-mom => hydrostatic
1 ∂p
0=−
−g
ρ ∂z
4
z
z=η(x,y,t)
6
η
p = pa + ∫ ρgdz
z
Use in x and y eqns
z
Specific terms, cont’d
Term 4
η
⎤
∂p ∂ ⎡
= ⎢ pa + ∫ ρgdz ⎥
∂x ∂x ⎢⎣
⎥⎦
z
z=η(x,y,t)
z
η
=
∂pa ∂η
∂ρ
ρ s g + ∫ gdz
+
∂x ∂x
∂x
z
4a
4b
4c
z
4a atmospheric pressure gradient (often negligible)
4b barotropic pressure gradient (barotropic => ρ = ρs = const)
4c baroclinic pressure gradient (baroclinic => density gradients;
often negligible)
Simplification
Neglect εxx; εxy = εh; εzx = εz
Neglect all pressure terms except barotropic
And drop over bars
∂η ∂ ⎛ ∂u ⎞ ∂ ⎛ ∂u ⎞
∂
∂
∂u ∂ 2
⎟⎟ + ⎜ ε z
+ ⎜⎜ ε h
+ (u ) + (uv) + (uw) − fv = − g
⎟
∂x ∂y ⎝ ∂y ⎠ ∂z ⎝ ∂z ⎠
∂z
∂y
∂t ∂x
1
2a
3
4b
2b1
2b2
Contrast with mass transport eqn
∂c ∂
∂
∂
+ (uc) + (vc) + ( wc)
∂t ∂x
∂y
∂z
1
2a
=
∂ ⎛ ∂c ⎞ ∂ ⎛ ∂c ⎞ ∂ ⎛ ∂c ⎞
⎜⎜ E x ⎟⎟ + ⎜⎜ E y ⎟⎟ + ⎜ E z ⎟ ± ∑ r
∂x ⎝ ∂y ⎠ ∂y ⎝ ∂y ⎠ ∂z ⎝ ∂z ⎠
2b1
2b2
7
Pressure gradient (4) in momentum eqn => viscosity (2b) not always
important to balance advection (2a); depends on shear (separation). For
mass transport, diffusivity (2b) always needed to balance advection (2a)
Comments
3D models include continuity + three
components of momentum eq (z may be
hydrostatic approx) + n mass transport eqns
Above are primitive eqs (u, v, w); sometimes
different form, but physics should be same
Sometimes further simplifications
Spatial averaging => reduced dimensions
Further possible simplifications
Neglect terms 2a and 2b1
∂u
∂η ∂ ⎛ ∂u ⎞
+ − fv = − g
+ ⎜ε z
⎟
∂t
∂x ∂z ⎝ ∂z ⎠
Linear shallow
water wave eqn
Also neglect term 1
∂η ∂ ⎛ ∂u ⎞
− fv = − g
+ ⎜ε z
⎟
∂x ∂z ⎝ ∂z ⎠
Steady Ekman flow
Also neglect term 2b2
∂η
− fv = − g
∂x
Geostrophic flow
Spatial averaging
z-vertical
y-lateral
3-D equations
φ(x,y,z,t) ocean
x-longitudinal
2-D vertical average
2-D lateral average
φ(x,y,t) shallow coastal;
estuary
φ(x,z,t) long reservoir; deep
estuary/fjord
1-D vertical & lateral
average
φ(x, t) river; narrow/shallow
estuary
1-D horizontal average
φ(z, t) deep lake/reservoir
ocean
Comments
Models of reduced dimension achieved by
spatial averaging or direct formulation
(advantages of both)
Demonstration of vertical averaging
(integrate over depth then divide by depth,
leading to 2D depth-averaged models)
Discussion of cross-sectional averaging (river
models)
Vertical Integration => 2D
(depth-averaged) eqns
z
H(x,y,t)
η(x,y,t)
y
h(x,y,t)
x
u ( x, y, z , t ) = u ( x, y, t ) + u" ( x, y, z , t )
u ( x , y , z , t ) = U h ( x , y , t ) + u" ( x , y , z , t )
(notes use)
Vertical Integration, cont’d
z
η
H
u
u"
c
c"
-h
1
u ( x, y , t ) =
H
η
∫ u ( x, y, z, t )dz
−h
In analogy with Reynolds averaging, decompose velocities and
concentrations into
u + u" , c + c"
, etc. and spatially average
Depth-averaged eqns
Continuity
∂η ∂
∂
+ (u H ) + (v H ) = 0
∂t ∂x
∂y
∂u ∂v
∂w
+ kinematic surface bc of
,
∂z
∂x ∂y
from
Mass and Momentum
η
∂ 2
∂
=
(
)
u
dz
∫ ∂x
∂x ∫
−h
=
( )
straight forward except for NL terms
2
2
∂
∂
u + u" u + u" dz = ∫ u dz + 2∫ u u" dz + ∫ u" dz
∂x
∂x
(
)(
)
∂ 2
∂
u H + ⎛⎜ u"2 H ⎞⎟
⎠
∂x
∂x ⎝
momentum dispersion
∂
∂
∂
=
+
(
uc
)
dz
u
c
H
u" c"H
∫−h ∂x
∂x
∂x
η
(
)
(
)
mass dispersion
Depth-averaged eqns, cont’d
x-momentum
∂
(u H ) + ∂ u 2 H + ∂ (u v H ) − fv H = − gH ∂η
∂t
∂x
∂y
∂x
(
+
)
∂ ⎡
∂u ⎤
H
ε
+
L
⎥
⎢
∂x ⎣
∂x ⎦
⎤
∂ ⎡
∂u
2
− u" H ⎥
⎢H ε x
∂x ⎢⎣
∂x
⎥⎦
Depth-ave
long. diff
∂ ⎡
∂v ⎤ τ sx τ bx
+
−
H
ε
T
⎥
⎢
ρ
∂y ⎣
∂x ⎦ ρ
⎤
∂ ⎡
∂u
− u" v"H ⎥
⎢H ε y
∂y ⎢⎣
∂y
⎥⎦
Long. dispersion
Depth integrated eqns, cont’d
Mass Transport
∂
(c H ) + ∂ (u c H ) + ∂ (v c H ) =
∂t
∂x
∂y
+
∂ ⎡
∂c ⎤
+
HE
L
∂x ⎢⎣
∂x ⎥⎦
⎤
∂ ⎡
∂c
− u" c"H ⎥
⎢H Ex
∂x ⎢⎣
∂x
⎥⎦
Depth-ave
long. diff
∂c
∂c
∂ ⎡
∂c ⎤
+
−
HE
E
E
T
z
z
∂z s
∂z
∂y ⎢⎣
∂y ⎥⎦
⎤
∂ ⎡
∂c
− v" c"H ⎥
⎢H E y
∂y ⎢⎣
∂y
⎥⎦
Long dispersion
b
Comments
εL, εT, EL , ET are longitudinal and transverse
momentum and mass shear
viscosity/dispersion coefficients.
εL >> εT and EL >> ET , but relative
importance depends on longitudinal gradients
Dispersion process represented as Fickian
(explained shortly)
Boundary Conditions: momentum
τ sx
2
2
= C D U w + Vw U w
ρ
Surface shear stress due to wind
(components Uw and Vw); external input
(unless coupled air-water model)
Assumes Uw >> us; CD = drag
coefficient ~ 10-3 (more in Ch 8)
τ bx
= Cf u 2 +v 2 u
ρ
≅ u*
2
Bottom shear stress caused by flow
(computed by model)
Different models for Cf (DarcyWeisbach f; Manning n; Chezy C),
e.g.,
τ bx
f
= ρu 2
8
Boundary Conditions: mass transport
z
∂c
Ez
∂z s
Surface mass transfer
(air-water exchange)
c
∂c
=0
∂z s
No flux
(dye, salt)
∂c
>0
∂z s
Source
(DO)
∂c
<0
∂z s
Sink
(VOC)
Boundary Conditions: mass transport
∂c
− Ez
∂z b
Benthic mass transfer
(sediment-water exchange)
z
c
−
∂c
=0
∂z b
No flux
(dye, salt)
−
∂c
>0
∂z b
Source (pore
water diffusion)
∂c
−
<0
∂z b
Sink (trace metals bound
by anoxic sediments)
Magnitude of terms: Ez
E z ~ u ' L ~ u* H
u* = shear velocity = τ b / ρ =
f
u,
8
f ≅ 0.02 => u* = 0.05u
∆x
A
Fg = A∆xρSg
S
u
= F f = τ b p∆x = ρu* p∆x
2
ASg
= R H Sg ; u* ≅ gHS ;
p
R H = A / p = hydraulic radius ≅ H
u* =
2
Normal flow; gravity
balances friction
Ez cont’d
E z = 0.07u* H
τ vm ≅
0.5 H 2
Ez
xvm = u τ vm
Seen previously; from analogy of mass
and momentum conservation (Reynolds’
analogy) and log profile for velocity
0.5 H 2
7H
≅
≅
0.07u* H
u*
7 Hu
≅
; if u = 20u*
u*
xvm ≅ 150 H
Transverse mixing: ET
ET
≅ 0.08 − 0.24
u* H
(say 0.15)
ET
≅ 0.2 − 4.6
u* H
(say 0.6)
τ tm
0.5B 2
0.5 B 2
≅
=
0.6u* H
ET
xtm
0.5B 2
=
u ; if u = 20u*
0.6u* H
17 B 2
=
H
Laboratory rectangular channels
Real channels (irregularities,
braiding, secondary circulation)
B = channel width
Example
B = 100 m, H = 5m, u = 1 m/s
Xvm = 150H = 750 m
xtm = 17B2/H = (17)(100)2/5 = 34,000 m
(34 km)
It may take quite a while before concentrations can be
considered laterally (transversally) uniform
Simplifications
Steady state; depth-averaged; no lateral advection or long dispersion;
no boundary fluxes
∂
(c H ) + ∂ (u c H ) + ∂ (v c H ) = ∂ ⎡⎢ HE L ∂c ⎤⎥ + ∂ ⎡⎢ HET ∂c ⎤⎥ + E z ∂c − E z ∂c
∂y ⎦
∂t
∂x
∂y
∂x ⎣
∂x ⎦ ∂y ⎣
∂z s
∂z b
Hu
∂c
∂ ⎛
∂c ⎞
⎟⎟
= ⎜⎜ HET
∂x ∂y ⎝
∂y ⎠
Uniform channel
∂c ET ∂ 2 c
=
∂x
u ∂y 2
const
Simple diffusion equation: solutions for
continuous source at x=y=0
y
B
u
x
y
B
qd - dimensionless cumulative
discharge
1.0
0.01
0.8
0.001
0.6
0.7
0.3
cd - dimensionless concentration
0.99
0.4
1=cd
0.2
1.1
1.3
3
30
0
0.0001
10
0.001
0.01
0.1
1.01
1
xET
uB2
1.6
Note:
1.2
qd = 0
xtm
0.8
qσ = 1
0.4
0
Holly and Jirka
(1986)
0.1
2.0
cBu H
m&
0.9
0
0.2
0.4
0.6
xd - dimensionless longitudinal distance
0.5 B 2 u
≅
ET
xET
uB2
Figure by MIT OCW.
A useful extension: cumulative
discharge approach (Yotsukura & Sayre, 1976)
Use cumulative discharge (Qc) instead of y as lateral variable
Q
y
Qc = ∫ H ( y ' )u ( y ' )dy '
Q
0
∂c
∂
=
∂x ∂Qc
Qc
⎡ 2
∂c ⎤
H
u
E
T
⎢
⎥
Q
∂
c ⎦
⎣
D
D behaves mathematically like
diffusion coefficient, but has different
dimensions; can be approximated as
constant (cross-sectional average):
Q
1
∂ 2c
∂c
2
D = ∫ H u ET dQc =>
= D
2
Q 0
∂x
∂Qc
c
u
H
Can use previous
analysis
y
Longitudinal Shear Dispersion
Why is longitudinal dispersion Fickian?
Original analysis by Taylor (1953, 1954) for flow in
pipes; following for 2D flow after Elder (1959)
z
u
u”
Longitudinal Shear dispersion
Why is longitudinal dispersion Fickian?
Original analysis by Taylor (1953, 1954) for flow in
pipes; following for 2D flow after Elder (1959)
z
H
u
to
u”
t
t1
c
t2
L2
t
Shear dispersion, cont’d
z
H
u”
u
t
to
t1
t2
L2
∂c ∂ ⎛ ∂c ⎞ ∂ ⎛ ∂c ⎞
∂c
+ u ( z ) = ⎜ E x ⎟ + ⎜⎜ E y ⎟⎟
∂x ∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ⎠
∂t
u = u + u"; c = c + c"; x = ζ + u t ; t = τ
∂c" ∂ ⎛ ∂c ⎞ ∂ ⎛ ∂c" ⎞ ∂ ⎛ ∂c ⎞ ∂ ⎛ ∂c" ⎞
∂c
∂c ∂c"
⎜⎜ E x
⎜⎜ E x
⎟⎟ +
⎟⎟ + ⎜ E z
=
+ u"
+ u"
+
⎟ + ⎜ Ez
⎟
∂ζ ∂ζ ⎝ ∂ζ ⎠ ∂ζ ⎝ ∂ζ ⎠ ∂z ⎝ ∂z ⎠ ∂z ⎝
∂ζ
∂z ⎠
∂τ ∂τ
1
2
3
4
5
6
7
Shear dispersion, cont’d
z
H
u”
u
t
to
t1
a ) L >> H
t2
L2
=> longitudinal dispersion << vertical
b) c >> c" => ∂c" ∂ζ << ∂c ∂ζ
5, 6 << 7
4 << 3
c) x >> L => ∂ ∂τ << u" ∂ ∂ζ
1, 2 << 3
∂c ∂c"
∂c
∂c" ∂ ⎛ ∂c ⎞ ∂ ⎛ ∂c" ⎞ ∂ ⎛ ∂c" ⎞
⎟⎟ + ⎜ E z
⎟⎟ +
⎜⎜ E x
⎜⎜ E x
+
+ u"
+ u"
=
⎟
∂τ ∂τ
∂ζ
∂z ⎠
∂ζ ∂ζ ⎝ ∂ζ ⎠ ∂ζ ⎝ ∂ζ ⎠ ∂z ⎝
1
2
3
4
5
6
7
Shear dispersion, cont’d
z
H
u
u”
t
to
t1
∂ ⎛ ∂c" ⎞
∂c
= ⎜ Ez
u"
⎟
∂z ⎠
∂ζ ∂z ⎝
3
We want
7
− u"c"
t2
L2
Differential advection balanced by
transverse diffusion
and to show that it is ~
∂c
EL
∂x
Integrate over z twice to get c”; multiply by u”; integrate again and divide
by H (depth average); add minus sign
Shear dispersion, cont’d
1
1 ∂c
1
u"∫ ∫ u" dzdzdz
− u" c" = − ∫ u" c" dz = −
∫
H 0
H ∂ζ 0 0 E z 0
H
EL = I h
H2
H
2
u"
Ez
z
z
EL!!!
Ih = dimensionless triple integration ~ 0.07
u" ~ u* ; E z ~ u* H
EL = 5.9u*H
Elder (1959) using log profile for u”(z)
EL = 10u*ro
Taylor (1954) turbulent pipe flow (ro= radius)
Use EL to compute σL or use measured σL to deduce EL
Comments
EL involves differential advection (u”)
with transverse mixing in direction of
advection gradient (Ez)
EL ~ 1/Ez; perhaps counter-intuitive, but
look at time scales:
EL ~
H
2
u"2
Ez
Tc Uc2
Recall Taylor’s
Theorem
∞
D ~ u 2 ∫ R(τ )dτ
0
A thought experiment
Consider the trip on the Mass Turnpike from Boston to
the NY border (~150 miles).
Assume two lanes in each direction, and that cars in
left lane always travel 65 mph, while those in the right
lane travel 55 mph.
At the start 50 cars in each lane have their tops
painted red and a helicopter observes the “dispersion”
in their position as they travel to NY
1) How does this dispersion depend on the frequency
of lane changes?
2) Would dispersion increase or decrease if there were
a third (middle) lane where cars traveled at 60 mph?
Thought experiment, cont’d
EL ~
H
2
(u" )
2
Ez
What are the analogs of H, Ez and u”
Thought experiment, cont’d
H2
(u" ) 2
EL ~
Ez
H ~ number of lanes
Ez ~ frequency of lane changing
u” ~ difference between average and lane-specific speed
1) Decreasing Ez increases dispersion (as long as there is some Ez)
2) Increasing lanes increases H, decreases mean square u”,
(65 − 60) 2 + (55 − 60) 2
(2 lanes)
= 50 / 2
(u" ) =
2
(65 − 60) 2 + (60 − 60) 2 + (55 − 60) 2
2
(u" ) =
= 50 / 3 (3 lanes)
3
2
If Ez is constant, net effect is increase in EL by (3/2)2(2/3) = 50%
1D (river) dispersion
u = u + u"; c = c + c"
c=
1
cdA
∫
AA
Insert into GE and spatial average
Continuity
∂A ∂
+ ( Au ) = q L
∂t ∂x
qL = lateral inflow/length [L2/T]
Mass Conservation (conservative form)
∂ ⎡
∂c
∂
∂
⎤
−
A
u
c
A
E
+ Ari + q L c L
( Ac) + ( Auc) =
"
"
x
⎢
⎥
∂x ⎣
∂x
∂x
∂t
⎦
AE L
∂c
∂x
Longitudinal dispersion again
1D (river) dispersion, cont’d
NC form from conservative equation minus c times continuity
∂
∂
∂ ⎡
∂c
⎤
( Ac) + ( Auc) =
−
"
"
A
E
A
u
c
x
⎥ + Ari + q L c L
∂t
∂x
∂x ⎢⎣
∂x
⎦
⎧ ∂A ∂
⎫
− c ⋅ ⎨ + ( Au ) = q L ⎬
⎩ ∂t ∂x
⎭
Mass Conservation (NC form)
qL
∂c
∂c 1 ∂ ⎛
∂c ⎞
(c L − c )
+u
AE
=
r
+
+
⎜ L ⎟ i
∂t
∂x A ∂x ⎝
∂x ⎠
A
Note: if cL > c, c increases;
if cL < c, c decreases (dilution)
EL for rivers
Elder formula accounts for vertical shear (OK for depth averaged
models that resolve lateral shear); here we need to parameterize
lateral and vertical shear. Analysis by Fischer (1967)
B 2 2 ∂c
− u" c" = I b
u"
ET
∂x
EL
B = river width
ET = transverse dispersion coefficient
Ib – ND triple integration (across A) ~ 0.07
Same form as Elder, but now time scale is B2/ET, rather than H2/Ez.
ET > Ez, but B2 >> H2 => this EL is generally much larger
EL for rivers, cont’d
Using approximations for u”, ET, etc.
or
u 2B2
E L ≅ 0.01
u* H
2
⎛u ⎞ ⎛B⎞
EL
≅ 0.01⎜⎜ ⎟⎟ ⎜ ⎟
u* H
⎝ u* ⎠ ⎝ H ⎠
if u ≅ 20u*
EL
⎛B⎞
≅ 4⎜ ⎟
u* H
⎝H⎠
2
Fischer (1967); useful for reasonably
straight, uniform rivers and channels
2
Magnitude of terms, revisited
Ez
≅ 0.07
u* H
Vertical Diffusion
ET
≅ 0.6
u* H
Transverse Diffusion in Channels
EL
≅6
u* H
Longitudinal Dispersion (depth-averaged flow)
EL
≅ 10
u* ro
Longitudinal Dispersion (turbulent pipe flow)
2
⎛u ⎞ ⎛B⎞
EL
≅ 0.01⎜⎜ ⎟⎟ ⎜ ⎟ Longitudinal Dispersion (rivers)
u* H
⎝ u* ⎠ ⎝ H ⎠
2
Previous Example, revisited
B = 100 m, H = 5m, u = 1 m/s, u* = 0.05u =0.05
0.01u 2 B 2 (0.01)(1) 2 (100) 2
EL =
=
u* H
(0.05)(5)
d
σ x 2 = 2E L
dt
xtm
= 400 m 2 /s
=> Gaussian Distribution; but only
after cross-sectional mixing
0 .5 B 2
u
=
ET
xtm = 34 km if point source on
river bank;
xtm
u
B
Previous Example, revisited
B = 100 m, H = 5m, u = 1 m/s, u* = 0.05u =0.05
0.01u 2 B 2 (0.01)(1) 2 (100) 2
EL =
=
u* H
(0.05)(5)
d
σ x 2 = 2E L
dt
xtm
= 400 m 2 /s
=> Gaussian Distribution; but only
after cross-sectional mixing
0 .5 B 2
u
=
ET
4 times less if point source in
mid-stream
xtm
u
B
Previous Example, revisited
B = 100 m, H = 5m, u = 1 m/s, u* = 0.05u =0.05
0.01u 2 B 2 (0.01)(1) 2 (100) 2
EL =
=
u* H
(0.05)(5)
d
σ x 2 = 2E L
dt
xtm
= 400 m 2 /s
=> Gaussian Distribution; but only
after cross-sectional mixing
0 .5 B 2
u
=
ET
Less still if distributed across
channel (but not zero)
xtm
u
B
Storage zones
Real channels often have backwater (storage) zones that
increase dispersion and give long tails to c(t) distribution
A
As
c(x,t)
t
Storage zones, cont’d
1) ∂c + u ∂c = 1 ∂ ⎛⎜ AE ∂c ⎞⎟ + q L (c − c ) + α (c − c)
L
L
s
∂t
∂x A ∂x ⎝
∂x ⎠ A
dc s
A
(c s − c )
=
−
α
2) dt
As
Main channel
Storage zone
A(x) = cross-sectional area of main channel
As = cross-sectional area of storage zone
α = storage zone coefficient (rate, t-1; like qL/A)
If you multiply 1) by A and 2) by As, the exchange
terms are αA(cs-c) and –αA(cs-c)
Really the same process as longitudinal dispersion, but instead of
cars in either fast or slow lane, some are in the rest stop.