Scale analysis of the toy model equations (Petrie & Bannister)

Ross Bannister, December 2016

For the characteristic values we set u = Uu^*, v = Vv^*, w = Ww^*, ρ̃’ = P’ρ̃’^*, ρ̃ ~ 1ρ̃^*, and b’ = ℬ’b’^*.
For the characteristic horizontal length-scales we set (respectively for each variable) x = ℒ^H_ux^*_u, x = ℒ^H_vx^*_v, x = ℒ^H_wx^*_w, x = ℒ^H_ρ̃’x^*_ρ̃’ and x = ℒ^H_b’x^*_b’.
For the vertical length-scales z = ℒ^V_uz^*_u, z = ℒ^V_vz^*_v, z = ℒ^V_wz^*_w, z = ℒ^V_ρ̃’z^*_ρ̃’ and z = ℒ^V_b’z^*_b’.
The timescale is set as t = [ℒ^H_u ⁄ (BU)]t^*.

The original equations:

(1) (∂u)/(∂t) + Bu(∂u)/(∂x) + Bw(∂u)/(∂z) + C(∂ρ̃’)/(∂x) − fv = 0, (2) (∂v)/(∂t) + Bu(∂v)/(∂x) + Bw(∂v)/(∂z) + fu = 0, (3) (∂w)/(∂t) + Bu(∂w)/(∂x) + Bw(∂w)/(∂z) + C(∂ρ̃’)/(∂z) − b’ = 0, (4) (∂ρ̃’)/(∂t) + B(∂ρ̃u)/(∂x) + B(∂ρ̃w)/(∂z) = 0, (5) (∂b’)/(∂t) + Bu(∂b’)/(∂x) + Bw(∂b’)/(∂z) + A²w = 0.

Introduce the change of variables:

(6) (BU²)/(ℒ^H_u)(∂u^*)/(∂t^*) + (BU²)/(ℒ^H_u)u^*(∂u^*)/(∂x^*_u) + (BUW)/(ℒ^V_u)w^*(∂u^*)/(∂z^*_u) + (CP’)/(ℒ^H_ρ̃’)(∂ρ̃’^*)/(∂x^*_ρ̃’) − fVv^* = 0, (7) (BUV)/(ℒ^H_u)(∂v^*)/(∂t^*) + (BUV)/(ℒ^H_v)u^*(∂v^*)/(∂x^*_v) + (BVW)/(ℒ^V_v)w^*(∂v^*)/(∂z^*_v) + fUu^* = 0, (8) (BUW)/(ℒ^H_u)(∂w^*)/(∂t^*) + (BUW)/(ℒ^H_w)(∂w^*)/(∂x^*_w) + (BW²)/(ℒ^V_w)w^*(∂w^*)/(∂z^*_w) + (CP’)/(ℒ^V_ρ̃’)(∂ρ̃’^*)/(∂z^*_ρ̃’) − ℬ’b’^* = 0, (9) (BUP’)/(ℒ^H_u)(∂ρ̃’^*)/(∂t^*) + (BUP’)/(ℒ^H_u)(∂ρ̃^*u^*)/(∂x^*_u) + (BWP’)/(ℒ^V_w)(∂ρ̃^*w^*)/(∂z^*_w) = 0, (10) (BUℬ’)/(ℒ^H_u)(∂b’^*)/(∂t^*) + (BUℬ’)/(ℒ^H_b’)u^*(∂b’^*)/(∂x^*_b’) + (BWℬ’)/(ℒ^V_b’)w^*(∂b’^*)/(∂z^*_b’) + A² Ww^* = 0.

Divide (6↑) by Vf; divide (6↑) by Wf; divide (6↑) and (6↑) by f, and introduce the dimensionless parameter Ro = U ⁄ (fℒ^H_u):

(11) BRo(∂u^*)/(∂t^*) + BRo u^*(∂u^*)/(∂x^*_u) + (BW)/(fℒ^V_u)w^*(∂u^*)/(∂z^*_u) + (CP’)/(Ufℒ^H_ρ̃’)(∂ρ̃’^*)/(∂x^*_ρ̃’) − (V)/(U)v^* = 0, (12) BRo(∂v^*)/(∂t^*) + (BU)/(fℒ^H_v)u^*(∂v^*)/(∂x^*_v) + (BW)/(fℒ^V_v)w^*(∂v^*)/(∂z^*_v) + (U)/(V)u^* = 0, (13) BRo(∂w^*)/(∂t^*) + (BU)/(fℒ^H_w)(∂w^*)/(∂x^*_w) + (BW)/(fℒ^V_w)w^*(∂w^*)/(∂z^*_w) + (CP’)/(Wfℒ^V_ρ̃’)(∂ρ̃’^*)/(∂z^*_ρ̃’) − (ℬ’)/(Wf)b’^* = 0, (14) BP’Ro(∂ρ̃’^*)/(∂t^*) + BP’Ro(∂ρ̃^*u^*)/(∂x^*_u) + (BP’W)/(fℒ^V_w)(∂ρ̃^*w^*)/(∂z^*_w) = 0, (15) Bℬ’Ro(∂b’^*)/(∂t^*) + (BUℬ’)/(fℒ^H_b’)u^*(∂b’^*)/(∂x^*_b’) + (BWℬ’)/(fℒ^V_b’)w^*(∂b’^*)/(∂z^*_b’) + (A² W)/(f)w^* = 0.

Introduce more ratios of speeds and length-scales to allow Ro to be used more frequently:

(ℒ^H_u)/(U)

(16) BRo(∂u^*)/(∂t^*) + BRo u^*(∂u^*)/(∂x^*_u) + B(ℒ^H_u)/(ℒ^V_u)(W)/(U)Ro w^*(∂u^*)/(∂z^*_u) + (CP’)/(Ufℒ^H_ρ̃’)(∂ρ̃’^*)/(∂x^*_ρ̃’) − (V)/(U)v^* = 0, (17) BRo(∂v^*)/(∂t^*) + B(ℒ^H_u)/(ℒ^H_v)Ro u^*(∂v^*)/(∂x^*_v) + B(ℒ^H_u)/(ℒ^V_v)(W)/(U)Ro w^*(∂v^*)/(∂z^*_v) + (U)/(V)u^* = 0, (18) BRo(∂w^*)/(∂t^*) + B(ℒ^H_u)/(ℒ^H_w)Ro(∂w^*)/(∂x^*_w) + B(ℒ^H_u)/(ℒ^V_w)(W)/(U)Ro w^*(∂w^*)/(∂z^*_w) + (CP’)/(Wfℒ^V_ρ̃’)(∂ρ̃’^*)/(∂z^*_ρ̃’) − (ℬ’)/(Wf)b’^* = 0, (19) BP’Ro(∂ρ̃’^*)/(∂t^*) + BP’Ro(∂ρ̃^*u^*)/(∂x^*_u) + BP’(ℒ^H_u)/(ℒ^V_w)(W)/(U)Ro(∂ρ̃^*w^*)/(∂z^*_w) = 0, (20) Bℬ’Ro(∂b’^*)/(∂t^*) + Bℬ’(ℒ^H_u)/(ℒ^H_b’)Ro u^*(∂b’^*)/(∂x^*_b’) + Bℬ’(ℒ^H_u)/(ℒ^V_b’)(W)/(U)Ro w^*(∂b’^*)/(∂z^*_b’) + (A² W)/(f)w^* = 0.

Let us introduce the following dimensionless parameters (and divide (16↑) by ℬ’):

A = aspect ratio for the zonal wind = ℒ^V_u ⁄ ℒ^H_u.
W_U = vertical to zonal wind = W ⁄ U.
V_U = meridional to zonal wind = V ⁄ U.

(21) BRo⎡⎣(∂u^*)/(∂t^*) + u^*(∂u^*)/(∂x^*_u) + A^− 1 W_Uw^*(∂u^*)/(∂z^*_u)⎤⎦ + (CP’)/(Ufℒ^H_ρ̃’)(∂ρ̃’^*)/(∂x^*_ρ̃’) − V_Uv^* = 0, (22) BRo⎡⎣(∂v^*)/(∂t^*) + (ℒ^H_u)/(ℒ^H_v)u^*(∂v^*)/(∂x^*_v) + (ℒ^H_u)/(ℒ^V_v)W_Uw^*(∂v^*)/(∂z^*_v)⎤⎦ + V^− 1_Uu^* = 0, (23) BRo⎡⎣(∂w^*)/(∂t^*) + (ℒ^H_u)/(ℒ^H_w)(∂w^*)/(∂x^*_w) + (ℒ^H_u)/(ℒ^V_w)W_Uw^*(∂w^*)/(∂z^*_w)⎤⎦ + (CP’)/(Wfℒ^V_ρ̃’)(∂ρ̃’^*)/(∂z^*_ρ̃’) − (ℬ’)/(Wf)b’^* = 0, (24) (∂ρ̃’^*)/(∂t^*) + (∂ρ̃^*u^*)/(∂x^*_u) + (ℒ^H_u)/(ℒ^V_w)W_U(∂ρ̃^*w^*)/(∂z^*_w) = 0, (25) BRo⎡⎣(∂b’^*)/(∂t^*) + (ℒ^H_u)/(ℒ^H_b’)u^*(∂b’^*)/(∂x^*_b’) + (ℒ^H_u)/(ℒ^V_b’)W_Uw^*(∂b’^*)/(∂z^*_b’)⎤⎦ + (A² W)/(ℬ’f)w^* = 0.

All terms are confirmed dimensionless.