Scale analysis of the toy model equations
RNB, 4th October 2014
The following scalings are used for the toy equations: u = Uu*, v = Vv*, w = Ww*, x = ℒx*, t = (ℒ ⁄ U)t*, z = ℋz*, ρ̃’ = P’ρ̃’*, ρ̃ = Pρ̃* and b’ = ℬ’b’*. In addition, the following definitions are used: Ro = U ⁄ (fℒ), M = U ⁄ √(BC), Au = U ⁄ V, Fr = U ⁄ √(gℋ), Aw = W ⁄ U, Ar = ℋ ⁄ ℒ.
RoM2Au⎡⎣(∂u*)/(∂t*) + Bu*(∂u*)/(∂x*) + BF2rAw(gℒ)/(U2)w*(∂u*)/(∂z*)⎤⎦ + RoAu(P’)/(B)(∂ρ̃’*)/(∂x*) − v*
=
0,
Ro⎡⎣(∂v*)/(∂t*) + Bu*(∂v*)/(∂x*) + B(Aw)/(Ar)w*(∂v*)/(∂z*)⎤⎦ + Auu*
=
0,
Ro⎡⎣(∂w*)/(∂t*) + Bu*(∂w*)/(∂x*) + (gℒ)/(U2)F2rAww*(∂w*)/(∂z*) + (P’)/(B)(1)/(AwAr)(1)/(M2)(∂ρ̃’*)/(∂z*)⎤⎦ − (ℬ’)/(fW)b’*
=
0,
( P’)/(P)(∂ρ̃’*)/(∂t*) + B(∂(ρ̃*u*))/(∂x*) + B(Aw)/(Ar)(∂(ρ̃*w*))/(∂z*)
=
0,
Ro⎡⎣(∂b’*)/(∂t*) + Bu*(∂b’*)/(∂x*) + B(Aw)/(Ar)w*(∂b’*)/(∂z*)⎤⎦ + (A2 W)/(ℬ’f)w*
=
0.
The scalings of each term are reproduced in the table below.
|
u1
|
u2
|
u3
|
u4
|
u5
|
|
RoM2Au
|
RoM2AuB
|
RoM2AuBF2rAwgℒ ⁄ U2
|
RoAuP’ ⁄ B
|
1
|
|
v1
|
v2
|
v3
|
v4
|
|
|
Ro
|
RoB
|
RoBAw ⁄ Ar
|
Au
|
|
|
w1
|
w2
|
w3
|
w4
|
w5
|
|
Ro
|
RoB
|
RogℒF2rAw ⁄ U2
|
RoP’ ⁄ (BAwArM2)
|
ℬ’ ⁄ (fW)
|
|
rhop1
|
rhop2
|
rhop3
|
|
|
|
P’ ⁄ P
|
B
|
BAw ⁄ Ar
|
|
|
|
bp1
|
bp2
|
bp3
|
bp4
|
|
|
Ro
|
RoB
|
RoBAw ⁄ Ar
|
A2 W ⁄ (ℬ’f)
|
|
In the above, the labels above each term allow each to be referenced easily.