6. GLOBAL WIND PATTERNS 92 thatrrot is larger than l. This is the case, for instance, on the sun, which is known to exhibit giant convection cells16. Then the diffusivity is justvcl, so the heat transported is
ε0 = vclvs2∆T
πR2T . (6.58)
Now suppose that l is the larger. Then ε0 = vcrrotvs2∆T
πR2T = v2cvs2∆T
2ΩπR2T cosθ. (6.59)
Now when ∆T ∼ T, it would appear that none of the above expansions are sufficient even qualitatively, for certain effects (such Richardson stabilization) do not appear to leading order. However, all of the models under consideration give in this case
v ∼ vs2
RΩ. (6.60)
In other words, the Mach number17 reduces to the sound speed Rossby number18. To get a feel for these Mach numbers, let us write R in units of R, Ω in units of 2π/1hour, and vs as 106T41/2cm/s, where T4 is the surface temperature measured in units of 104K. Note that we use a sound speed which is a factor of a few higher than that corresponding to 104K, as the temperature in the regions which transport significant heat by sonic winds is typically somewhere between a factor of one and ten higher than that at the surface. Using these values, we find that
v
vs ∼ T41/2Phour
50(R/R), (6.61)
where Phour is the orbital period measured in hours. As a result, we see that only for very short orbital periods can ∆T /T be of order unity with v/vs not of the same order. If such cases arise and are of interest, they may be handled by extrapolating the scaling with ∆T /T to the point wherev/vs is of order unity. We expect to incur minimal error by doing this, as the dynamic range of this scaling is at most 50.
6. GLOBAL WIND PATTERNS 93 Using the definition of each wavenumber we find
Ω3 R3ε
!1/5
>(3Ck)3/2
sλ3
ε . (6.62)
Now making use of
E˙ =ε−λE (6.63)
and
E = 1
2v2φ, (6.64)
we find that in steady-state
ε= 1
2λvφ2, (6.65)
and hence our condition is
2Ω3vφ3 R3λ
!1/5
>(3Ck)3/2√
2λ. (6.66)
The Rossby number for flow around the star is roughly Ro = vφ
2πRΩ. (6.67)
Using this we may writevφ= 2πRΩRo, such that 16π3Ω6
λ6
!1/5
Ro3/5 >(3Ck)3/2√
2. (6.68)
Evaluating the numerical constants yields roughly Ro>100 λ
Ω
!2
. (6.69)
Intuitively what this means is that the more the Coriolis force deflects the wind as it travels around the star, the faster the star needs to dissipate the winds in order to prevent bands from forming.
It is now worth examining how to compute the various quantities mentioned in discussing the Rhines formalism. Many of them have simple definitions but are nontrivial to arrive at from the externally specified fluid parameters, and so this is a somewhat tricky procedure.
6. GLOBAL WIND PATTERNS 94 To begin with then, consider λ. This may be interpreted as the timescale over which a wind dies down due to drag effects. Given that the Rhines cascade uses a quasi two-dimensional flow, the characteristic scale for the associated sheer will be the pressure scale height, and soλ may be estimated as
λ = E˙
E ≈ νvvφ2/h2s vφ2 = νv
h2s, (6.70)
where νv is the effective vertical viscosity on length scales of hs. Note that we neglect the viscosity in the horizontal direction, as this is already accommodated by the formalism of the Rhines arrest.
In the convection zone,νv =lmax (vc, vφ), so λ= ℵ
hsmax (vc, vφ). (6.71)
In the radiation zone, on the other hand,νh =v/kR, and so νv = v2φ(α+vφ/kR)
glℵ(∇ad− ∇) = vφ2α+qv3φR/Ω
glℵ(∇ad− ∇) , (6.72)
∴λ = v2φα+qvφ3R/Ω
gl2hs(∇ad− ∇) . (6.73)
The next quantity of interest isε. This is distinct from the ε used in the previous section, for here it is the power driving the wind, rather than the power the wind moves. Neglecting external heat input, in a steady state this will be the power lost by turbulence to drag, which is given by ˙E. This may be computed as in the previous paragraph. When external heat is included, however, some fraction of it should be counted towards this quantity. As discussed in Chapter 3, much of the external heating goes towards inducing a divergence in the flux. To compute the amount that goes towardsε, we use the same method as before, computing a power balance between the work extracted by the wind and the losses to bottom drag. The work extracted is, as usual,
W˙ = v2svφ∆T2
2πRT2 . (6.74)
The power lost is
E˙ =λE =ε= 1
2vφ2λ, (6.75)
6. GLOBAL WIND PATTERNS 95 where in the second equality we have assumed that the wind is in power equilibrium.
Setting ˙E equal to ˙W yields 1
2vφ2λ = v2svφ∆T2
2πRT2 (6.76)
∴vφλ = v2s∆T2
πRT2 . (6.77)
In the radiation zone this means that vφ3α+qvφ3R/Ω
gl2hs(∇ad− ∇) = vs2∆T2
πRT2 . (6.78)
A series expansion of this around ∆T /T = 0 yields vφ= vs2∆T2gl2hs(∇ad − ∇)
πRT2α
!1/3
. (6.79)
This may be simplified by noting that α= k
ρcp =− F
ρcp∂rT = F
ρ2gcp∂PT = F P
ρ2gcpT∇R = F hs
ρcpT∇R ≈ F hs
P∇R. (6.80) In the thin-shell approximation, we may write P = Σg and find
α= F hs
Σg∇R. (6.81)
6. GLOBAL WIND PATTERNS 96 Substituting this into the equation for vφ yields
vφ3 = vs2∆T2gl2hs(∇ad− ∇)
πRT2α (6.82)
= vs2∆T2g2Σ∇Rl2hs(∇ad− ∇)
πRT2F hs (6.83)
= vs2∆T2g2Σ∇Rl2(∇ad− ∇)
πRF T2 (6.84)
= vs2∆T2ℵ2P2Σ∇R(∇ad− ∇)
πRρ2F T2 (6.85)
= vs6∆T2ℵ2Σ∇R(∇ad− ∇)
πRγ2F T2 (6.86)
= vs6∆T2ℵ2Σ∇R(∇ad− ∇R)
πRγ2F T2 (6.87)
≈ v6s∆T2Σ∇R
πRF T2 , (6.88)
(6.89) where in the last line we have dropped some dimensionless constants of order unity.
As a result, we may write
λ= v2φα+qvφ3R/Ω
gl2hs(∇ad− ∇) (6.90)
= vs6∆T2Σ∇R πRF T2
!2/3
α+qvφ3R/Ω
gl2hs(∇ad− ∇) (6.91)
≈ v6s∆T2Σ∇R πRF T2
!2/3
α+qvφ3R/Ω
gh3s (6.92)
≈vs2 ∆T2Σ∇R πRF T2
!2/3
α+qvφ3R/Ω
h2s (6.93)
≈vs2 ∆T2Σ∇R
πRF T2
!2/3
F hs
Σg∇R +
qv6s∆T2Σ∇R
πΩF T2
h2s . (6.94)
(6.95) When ∆T /T is small, this simplifies to
λ= F∆T4 π2R2ΣT4∇R
!1/3
. (6.96)
6. GLOBAL WIND PATTERNS 97 The criterion for the Rhines scale to be in effect is then
Ro>100 λ Ω
!2
(6.97)
∴ vφ
2πRΩ >100 F∆T4 π2R2ΣT4∇RΩ3
!2/3
(6.98)
∴ 1 2πRΩ
v6s∆T2Σ∇R πRF T2
!1/3
>100 F∆T4 π2R2ΣT4∇RΩ3
!2/3
(6.99)
∴ 1 8π3R3Ω3
vs6∆T2Σ∇R πRF T2
!
>106 F2∆T8
π4R4Σ2T8∇2RΩ6 (6.100)
∴ 1 8Ω3
vs6∆T2Σ∇R F T2
!
>106 F2∆T8
Σ2T8∇2RΩ6 (6.101)
∴ 1 8
v6s∇R F
!
>106 F2∆T6
Σ3T6∇2RΩ3 (6.102)
∴vs6 >107 F3∆T6
Σ3T6∇3RΩ3 (6.103)
∴vs2 >100 F∆T2
ΣT2∇RΩ (6.104)
∴T4 >10−3 F∆T2Σh
FT2Σ∇RΩ (6.105)
∴T4 >100 F∆T2Σh FT2ΣΩ−4
. (6.106)
When it is in effect, the heat transported is ε0 =cpvφ∆T
πR ≈ 1
πRv2svφ∆T
T =vs3 16v3sΣ∇R l4F
!1/3
l 2πR
!4/3
∆T T
!5/3
. (6.107)
6. GLOBAL WIND PATTERNS 98 Recalling the definition of∇R, this becomes
ε0 =vs3 16vs3Σ∇R l4F
!1/3
l 2πR
!4/3
∆T T
!5/3
(6.108)
=vs3 16vs3Σ3κLP 16πacGM T4l4F
!1/3
l 2πR
!4/3
∆T T
!5/3
(6.109)
= v3s πR
2vs3Σ3κLP 16πacGM T4lF
!1/3
l 2πR
!1/3
∆T T
!5/3
(6.110)
= v3s πR
2vs3Σ3κLP 16πacgR2T4lF
!1/3
l 2πR
!1/3
∆T T
!5/3
(6.111)
= v3s πR
2vs3Σ3κ4πR2P 16πacgR2T4l
!1/3
l 2πR
!1/3
∆T T
!5/3
(6.112)
= v3s πR
2vs3Σ3κP 4acgT4l
!1/3
l 2πR
!1/3
∆T T
!5/3
(6.113)
= v3s πR
3vs3ΣκP 8σgT4l
!1/3
l 2πR
!1/3
∆T T
!5/3
(6.114)
= v3s πR
3vsΣκP γ 8σT4ℵ
!1/3
l 2πR
!1/3
∆T T
!5/3
. (6.115)
(6.116) When ∆T /T is large, on the other hand,
vφ
vs = v3s∆T2Σ∇R πRF T2
!1/3
∼
T43/2ΣFR∆T2 ΣhF RT2
1/3
, (6.117)
so we expect vφ to be of order vs. Note that if this formula indicates a speed greater than the sound speed we truncate it as usual to the sound speed. Using vφ ∼vs, we
6. GLOBAL WIND PATTERNS 99 find
λ= v2φα+qvφ3R/Ω
gl2hs(∇ad− ∇) (6.118)
≈ vφ2α+qv3φR/Ω
vs2h2s (6.119)
≈ vs2α+qvs3R/Ω
vs2h2s (6.120)
≈h−2s
α+
svs3R Ω
(6.121)
≈h−2s
F hs P∇R
+
svs3R Ω
(6.122)
≈h−2s 109 F hsΣhg
107cmFΣg + 1016T43/4
s
Ω−1−4 R R
!
(6.123)
≈
v u u t
vs3R
h4sΩ. (6.124)
The criterion for the Rhines scaling is then vs
2πRΩ >100 vs3R
h4sΩ3 ∴1>200πvs2R2
h4sΩ2 ∴1>200πg4R2
vs6Ω2. (6.125) As a rough estimate, the right side should be 109 or so for a sun-like star with Ω = 10−4s−1, so this case does not concern us.
We may now perform the same procedure for convecting regions, where vφℵ
hsmax (vc, vφ) = vs2∆T2
πRT2 . (6.126)
To solve this, we first assume vc> vφ and write vφ = hsvs2∆T2
vcℵπRT2. (6.127)
If this exceeds vc, then we instead use vφ=
shsv2s∆T2
ℵπRT2 . (6.128)
6. GLOBAL WIND PATTERNS 100 We then have
ε0 = vφcp∆T
πR . (6.129)
Once more λ and ε may be computed from these results. If vc> vφ, λ is a constant and ε goes as ∆T4. Otherwise,λ goes as ∆T and ε goes as ∆T3.