Download Scaling behavior in Rayleigh-Bénard convection

Scaling behavior in Rayleigh-Bénard convection with and
without rotation
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King, Eric; UC Berkeley, Earth and Planetary Science
Stellmach, Stephan; Westfälische Wilhelms-Universität Münster, Institut für
Geophysik
Buffett, Bruce; UC Berkeley, Earth and Planetary Science
Bénard convection < Convection, Rotating flows < Geophysical and
Geological Flows
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1
Under consideration for publication in J. Fluid Mech.
Scaling behavior in Rayleigh-Bénard
convection with and without rotation
E. M. K I N G1 , S. S T E L L M A C H2 ,
AND
B. B U F F E T T1
1
Department of Earth and Planetary Science, University of California, Berkeley, 94720-4767
USA
2
Institut für Geophysik, WWU Münster, AG Geodynamik Corrensstr. 24, Münster, 48149
Germany
3
Department of Earth and Space Sciences, University of California, Los Angeles, 90095-1567
USA
(Received soon)
Rotating Rayleigh-Bénard convection provides a simplified dynamical analog for many
planetary and stellar fluid systems. Here, we use numerical simulations of rotating RayleighBénard convection to investigate the scaling behavior of five quantities over a range of
Rayleigh (103 . Ra . 109 ), Prandtl (1 ≤ P r ≤ 100), and Ekman (10−6 ≤ E ≤ ∞). The
five quantities of interest are the viscous and thermal boundary layer thicknesses, mean
temperature gradients, typical horizontal length scales, and flow speeds (Peclet number).
We focus, where possible, on scalings with some theoretical basis. Also, the parameter
ranges in which different scalings apply are quantified, and so separate dynamical regimes
are identified.
Key Words: rotating convection
1. Introduction
Convection is a ubiquitous natural process, as fluids throughout the universe flow to
redistribute destabilizing thermal and gravitational energy stores. Astro- and geophysical bodies also rotate, and the convective flows in many of these systems are strongly
influenced by Coriolis acceleration.
We examine a simplified analog of these systems by way of Rayleigh-Bénard convection
(RBC) in a Boussinesq fluid with and without rotation. RBC consists of a fluid layer
sandwiched between two rigid, horizontal boundaries (Bénard 1900; Rayleigh 1916). The
bottom boundary is warmer than the top, and the gravity vector points downward. This
temperature drop, ∆T , destabilizes the fluid layer, as cool, heavier fluid lies atop warmer,
more buoyant fluid. The convection is treated as Boussinesq in that fluid properties are
isotropic and fixed in time and space, except in the buoyancy force, which is driven by
thermal expansion. For rotating RBC, the fluid layer is rotated about a vertical axis with
constant angular frequency Ω.
Rotating RBC in a Boussinesq fluid is governed by the conservation of momentum,
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E. M. King, S. Stellmach, and B. Buffett
Parameter
Definition
Simulations
Rayleigh Number Ra ≡ (αg∆T L3 )/(νκ) 103 . Ra . 109
Prognostic
Output
Ekman Number
E ≡ ν/(2ΩL2 )
Prandtl Number
P r ≡ ν/κ
1 ≤ P r ≤ 100
Nusselt Number
N u ≡ (qL)/(k∆T )
1 . N u . 50
Peclet Number
P e ≡ (U L)/κ
10−6 ≤ E ≤ ∞
3 < P e < 3000
Table 1. Relevant non-dimensional parameters. Dimensional quantities are as follows: α is the
fluid’s coefficient of thermal expansivity; g is gravitational acceleration; ∆T is the temperature
drop across the fluid layer; L is the depth of the fluid layer; ν is the fluid’s viscous diffusivity;
κ is the thermal diffusivity of the fluid; Ω is the angular rotation rate; q is mean heat flux; and
k is the fluid’s thermal conductivity.
mass, and internal energy (heat) (e.g., Chandrasekhar 1953):
1
∂u
(1.1)
+ u · ∇u + 2Ω × u = − ∇p + αgT + ν∇2 u ;
∂t
ρ0
∇·u= 0 ;
(1.2)
∂T
+ u · ∇T = κ∇2 T ;
(1.3)
∂t
respectively, where u, p, and T are velocity, modified pressure, and temperature fields,
respectively, ρ0 is the fluid’s mean density, and ν and κ are the viscous and thermal
diffusivity of the fluid.
There are seven global input parameters in the governing equations: Ω, ρ0 , α, g, ∆T, ν,
and κ; which contain four fundamental dimensions: time, mass, length, and temperature.
By Buckingham’s Π-theorem, the system is therefore defined by three independent prognostic parameters (Barenblatt 2003). Although there are an infinite number of ways to
combine these dimensional quantities into non-dimensional parameters, the fundamental
control parameters are usually taken to be Ra, E, and P r, which are defined in table 1.
The buoyancy forcing is characterized by the Rayleigh number, Ra. The period of rotation is characterized by the Ekman number, E. The Prandtl number, P r, characterizes
the fluid itself as a ratio of viscous to thermal diffusivities.
Rotating convection is generally found to occur in one of two basic regimes, geostrophic
or weakly rotating, typically as evidenced by heat transfer behavior (e.g., Schmitz &
Tilgner 2010). Heat transfer by geostrophic convection is suppressed relative to that by
convection without rotation (e.g., Rossby 1969; Liu & Ecke 2009), and has been argued
to scale as N u ∼ Ra3 E 4 using marginal boundary layer stability analysis (King et al.
2012). Heat transfer by weakly rotating convection is observed to scale similarly to that
by convection without rotation (e.g., Liu & Ecke 1997; Niemela et al. 2010). In the
range of Ra and P r accessed in this study, non-rotating convective heat transfer scales
as N u ∼ Ra2/7 (e.g., Chilla et al. 1993). As Ra increases, though, RBC studies observe
that this scaling exponent approaches the classical 1/3 value (e.g., Niemela & Sreenivasan
2006) predicted by the scaling analysis of ?.
Recent work has argued that the transition between the geostrophic and weakly rotating heat transfer regimes is related to the relative thicknesses of the thermal and Ekman
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Rotating convection scalings
Regime
Geostrophic
Weakly Rotating
Non-rotating
Criterion
RaE 3/2 < 10
RaE 3/2 > 10
E≈∞
Heat Transfer
N u = (Ra/Rac )3
N u = 0.16Ra2/7 N u = 0.16Ra2/7
Table 2. A summary of the results of King et al. (2012). The critical Rayleigh
number for rotating convection is Rac ≈ 7.6E −4/3 (Chandrasekhar 1953), such that
N u = (Ra/Rac )3 ≈ 0.0023Ra3 E 4 . We stress that although the geostrophic scaling law follows from boundary layer stability analysis, the weakly and non-rotating 2/7 law is empirical,
and probably a function of Ra and P r (e.g. Ahlers et al. 2009).
boundary layers (King et al. 2009, 2010). Geostrophic convection, they argue, should occur when the Ekman boundary layer is thinner than the thermal boundary layer. When
instead the thermal boundary layer is thinner, the influence of rotation should be weak.
The two boundary layers are observed to cross when RaE 3/2 ≈ 10 in numerical simulations of rotating convection. Transitions between the two heat transfer regimes are
observed to occur near this boundary layer transition in both numerical simulations and
laboratory experiments (King et al. 2012). These results are summarized in table 2.
Here, we extend this framework to investigate the scaling behavior of five separate
quantities in geostrophic (RaE 3/2 . 10), weakly rotating (10 . RaE 3/2 < ∞), and nonrotating (E = ∞) RBC simulations. The five quantities of interest are: viscous boundary
layer thickness, δv (section 3); thermal boundary layer thickness, δT (section 4); prevailing
thermal gradients, β (section 5); typical horizontal length scale of flow, ℓ (section 6); and
typical flow speed, P e (section 7).
2. Methods
2.1. Numerical Model
Time evolution of the velocity, pressure, and temperature fields are calculated as numerical solutions of the governing equations (1.1-1.3) in non-dimensional form:
E
Pr
∂u′
+ u′ · ∇u′
∂t
+ ẑ × u′ = −∇P + RaE T ′ ẑ + +E ∇2 u′
∇ · u′ = 0
(2.1)
(2.2)
∂T
+ u′ · ∇T ′ = ∇2 T ′
(2.3)
∂t
where u′ , P , and T ′ are the dimensionless velocity, modified pressure, and temperature,
respectively, and ẑ is the vertical unit vector. The fundamental length scale, L, is taken
to be the height of the fluid layer. The fundamental time scale is taken to be a thermal
diffusion time scale, L/κ2 , where κ is the fluid’s thermal diffusivity. The temperature
field is normalized by the temperature drop imposed across the layer, ∆T .
The computational domain is a cartesian box with periodic sidewalls (in order to approximate an infinite plane layer); rigid, no-slip top and bottom boundaries; and diameter to height aspect ratio, 1 ≤ Γ ≤ 4. Temporal discretization is accomplished through
a second-order, semi-implicit Adams Bashforth backward differentiation time stepping.
′
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800
L
E. M. King, S. Stellmach, and B. Buffett
Mean Horizontal Flow Speed
600
400
200
0
Height
3L/4
L/2
Trms
UH
T
L/4
0
0
ΔT/4
ΔT/2
Temperature
3 ΔT/4
ΔT
Figure 1. An example of profiles of mean temperature (hT ′ iH , solid line), temperature vari′
′
ance (Trms
, solid-filled curve), and mean horizontal flow speed (UH
, stripe-filled curve), from
a non-rotating convection simulation with P r = 1 and Ra = 7 × 105 . The horizontal dotted
and dashed lines depict the calculated thickness of the thermal and viscous boundary layers,
respectively.
Fourier series are used for spatial discretization in the horizontal direction, while Chebyshev polynomials are used in the vertical direction in order to better resolve the thin fluid
boundary layers. Ranges of input parameters accessed are given in Table 1. For further
details on the numerical method and validation, please see Stellmach & Hansen (2008)
and King et al. (2012).
2.2. Measurement Technique
We define the following nomenclature for averaging: an overbar, · · ·, represents time averaging; angled brackets represent spatial averaging over the entire computational domain,
h· · ·i, and over horizontal planes, h· · ·iH .
Viscous boundary layer thicknesses are calculated as the mean vertical distances of
the first local maximum of the horizontally averaged velocity variance above (below) the
bottom (top) domain boundary. That is, a vertical profile for the mean magnitude of the
horizontal velocity is acquired:
1/2
′
UH
(z) ≡ hu′H · u′H iH ;
(2.4)
and the local peak of this profile nearest each of the top and bottom boundaries represent
the edge of the top and bottom viscous boundary layers (Belmonte et al. 1994). An
example profile is shown in Figure 1. The top and bottom layer thicknesses are averaged,
giving the mean viscous boundary layer thickness, δv .
Thermal boundary layer thicknesses are similarly calculated, using vertical profiles for
temporal thermal variance,
′
Trms
(z) ≡ hT ′ − hT ′ iH iH .
(2.5)
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Rotating convection scalings
4
10
Peclet Number
Ekman Number
3
10
-3
10
-4
10
-5
10
-6
10
2
10
Prandtl Number
1
7
100
1
10
0
10 3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
Ra yleigh Number
Figure 2. Peclet number versus Rayleigh number, with Ekman numbers color-coded (non-rotating convection data are black), and Prandtl numbers are indicated by symbol size. This figure
illustrates the parameter range accessed by numerical simulations.
Again, local peaks in this temperature fluctuation profile give estimates for the locations
of the edges of the top and bottom thermal boundary layers, which are averaged to
calculate δT . An example of these profiles is shown in Figure 1.
The typical horizontal length scale of the flow, ℓ, is defined in the following way. Let
′
∗
E(k, m) = [u′(k,m) u(k,m)
] be the dimensionless kinetic energy contained in horizontal
modes k and m. The dominant horizontal scale can be determined as


P(kmax ,mmax )
2π (k=1,m=1)
E(k, m)
,
(2.6)
ℓ ≡  P(k ,m )
√
max
max
2 + m2
E(k,
m)
k
(k=1,m=1)
essentially calculating the mean wavelength of flow.
The Nusselt number characterizes the heat transport as
N u ≡ hu′z T ′ − ∂T ′ /∂zi,
(2.7)
and the Peclet number is defined as the typical amplitude of flow speed:
P e ≡ h(u′ · u′ )i1/2 .
(2.8)
Peclet number calculations are shown in Figure 2 plotted versus Ra to illustrate the
parameter range accessed in this study. Ekman numbers are denoted by symbol shape
(color online), and Prandtl numbers are denoted by symbol size.
Statistical f-tests are used to distinguish quality-of-fit between different scalings for
identical data. An f-test compares residual variance of data normalized by the two different scaling laws in question, and tests this comparison against the null hypothesis
that the two residuals populations have equal variance to within 5% significance. That
is, the ratio of the residual variances from the two scalings is compared with 95% confi-
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E. M. King, S. Stellmach, and B. Buffett
0.3
a
b
−1
δv /L
δv /L
10
0.05 0
10
−2
10
0.1
−3
1
10
2
3
10
10
10
−6
10
Reynolds Number
−5
10
−4
10
−3
10
Ekman Number
Figure 3. Symbols have the same meaning as in Figure 2. a) Viscous boundary layer thickness
versus the Reynolds number, Re = P e/P r, for simulations without rotation (E = ∞). The
solid line illustrates the slope of the predicted scaling δv ∼ Re−1/2 L. The dashed line shows
δv = 0.34Re−1/4 L. b) viscous boundary layer thickness versus the Ekman number, for rotating
convection simulations. Symbols have the same meaning as in Figure 2. The solid black line
shows δv = 3E 1/2 L.
dence bounds from an f-distribution with the same degrees of freedom as these residual
populations (Snedecor & Cochran 1980).
3. Viscous boundary layers, δv
3.1. Non-rotating
Viscous boundary layers in turbulent flows are usually assumed to be of the Blasius type,
derived by balancing the viscous term, which becomes important near the boundary, with
the inertial term that is thought to dominate in the bulk (e.g., Kundu 1990). Matching
these terms at the edge of the boundary layer gives:
u · ∇u ∼ ν∇2 u
→
U 2 /L ∼ νU/δv2 .
The viscous boundary layer thickness, δv , should then scale as
p
δv ∼ ν/U 2 L = Re−1/2 L,
(3.1)
(3.2)
where Re = P e/P r is the Reynolds number. This boundary layer scaling represents the
distance over which viscosity and acceleration by Reynolds stresses act on similar time
scales.
Figure 3a shows the thicknesses of the viscous boundary layer versus the Reynolds
number from non-rotating convection simulations. The solid line shows the slope of the
predicted scaling (3.2). The boundary layers within the simulations, however, are less
strongly dependent on Re. A best fit power law regression to these data with Re > 10
gives
δv = 0.34(±0.05)Re−0.25(±0.03)L.
(3.3)
This empirical scaling law is shown as the dashed line in Figure 3a, and agrees with the
data to within 5% for Re > 10. It is not currently clear why the non-rotating viscous
boundary layer scales weakly with Re, but this result is consistent with the extensive
experimental study of Lam et al. (2002), and the numerical simulations of Breuer et al.
(2004). We revisit this issue in the discussion section.
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Rotating convection scalings
7
3.2. Rotating
The viscous boundary layer in rotating flows is known as an Ekman boundary layer,
named after its discoverer V. W. Ekman (1905). The scaling behavior for the thickness
of the Ekman layer comes from a balance between the Coriolis term and the viscous term
near the top and bottom domain boundaries:
2Ω × u ∼ ν∇2 u
→
2U Ω ∼ νU/δv2 .
The viscous boundary layer thickness, δv , should then scale as
p
δv ∼ ν/2ΩL2 = E 1/2 L
(3.4)
(3.5)
in rotating convection. This boundary layer scaling represents the distance over which
viscosity acts in the course of one rotation period.
Figure 3b shows calculations of the thickness of the viscous boundary layer versus E for
rotating convection. Our most rapidly rotating data appear to conform to a δv /L ∝ E 1/2
(solid line). Indeed, we have twenty-six data points for rapidly rotating (E ≤ 10−4 )
geostrophic (RaE 3/2 < 10) convection, and a least-squares power law regression to these
data yields δv /L = 2.7(±0.3) E 0.49(±0.01). To within the uncertainty, this scaling can be
written as
δv /L = 3E 1/2 ,
(3.6)
which is shown as the solid line in Figure 3b. This scaling law fits the data to within 25%
on average for all rotating cases (E ≤ 10−3 ), and to within 8% on average for E ≤ 10−4 .
4. Thermal boundary layers, δT
4.1. Non-rotating
Turbulent convection, in the absence of constraining influences such as rotation, tends to
mix the bulk fluid. Temperature within the interior becomes uniform as Ra is increased.
The imposed temperature drop across the fluid layer is therefore accomplished almost
entirely within the thermal boundary layers. The Nusselt number, N u, is defined as the
ratio of total heat transport to that by conduction alone. Conductive heat transport
across the layer is given by qcond = k∆T /L. The total amount of heat transported by the
fluid can be determined, in this idealized case, by the thickness of the boundary layer.
Within the thermal boundary layer, heat transport is nearly entirely conductive, and so
qtotal ≈ k∆T /2δT . Thus, for turbulent, non-rotating convection, we expect (e.g., Spiegel
1971)
1
δT ≈ N u−1 L.
(4.1)
2
Figure 4 shows the thermal boundary layer thicknesses plotted versus N u (non-rotating
convection data are shown as circles). The solid line shows the predicted scaling (4.1),
which fits the non-rotating data to within 15% for all cases, and to within 5% for N u > 5
(Ra > 105 ).
4.2. Rotating
In figure 4, we observe that the thermal boundary layer thickness in rotating convection
is less well described by (4.1) than for convection without rotation. The quality of fit
of the thermal boundary layer thickness calculations to usual N u−1 scaling is especially
poor within the geostrophic regime. To see this, we plot in figure 5a (δT /L)N u versus
the parameter, RaE 3/2 , which was previously defined to identify the transition between
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E. M. King, S. Stellmach, and B. Buffett
0
δT /L
10
−1
10
−2
10
1
10
60
Nusselt Number
Figure 4. Thermal boundary layer thickness versus the Nusselt number for convection with
and without rotation. Symbols have the same meaning as in figure 2. The solid line shows the
slope of the predicted scaling (4.1), δT = L/(2N u).
geostrophic and weakly rotating regimes. The solid horizontal line shows the predicted
scaling for well-mixed turbulence (4.1), and the dotted horizontal lines indicate the standard deviation of non-rotating convection data about this prediction. In the geostrophic
regime (RaE 3/2 < 10) the misfit between thermal boundary layer thickness data and the
classical prediction (4.1) (solid horizontal line) is 52% on average, and can be as high as
170%. For weakly rotating convection (RaE 3/2 > 10), however, the thermal boundary
layer thickness data are fit by (4.1) with an average error of 13% with a peak misfit of
31%.
The poor agreement between (4.1) and geostrophic convection data is likely due to the
breakdown of the assumption that the interior fluid is well mixed when a strong rotational
influence is present. This affect of the Coriolis force has been previously observed in both
experiments (e.g., Boubnov & Golitsyn 1990) and simulations (e.g., Julien et al. 1996).
King et al. (2010) show that thermal mixing in geodynamo models can be subdued by
rotation for increasingly higher N u as E is decreased. They argue that as the Ekman layer
becomes thinner than the thermal boundary layer, the interior fluid maintains significant
thermal gradients, invalidating the classical thermal boundary layer thickness scaling
(4.1).
In figure 5b, we plot the relative thicknesses of the thermal and Ekman boundary layers,
δT /δv , calculated for all rotating convection simulations versus our predicted transition
RaE 3/2 . The boundary layers swap relative positions where δT /δv crosses unity, which
occurs somewhere in the range 7 < RaE 3/2 < 21 for all Ekman and Prandtl numbers
considered. We approximate this transition as RaE 3/2 ≈ 10, shown as the dashed vertical
line in figure 5.
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Rotating convection scalings
2
1
10
a
0.8
b
0.7
δT / δv
δT /L N u
1
10
0.6
0.5
0.4
0
10
0.3
0.2
0.1
−1
0
1
10
10
2
RaE 3/ 2
3
10
10
0
1
10
10
2
10
RaE
3
10
10
3/2
Figure 5. a) Thermal boundary layer thickness normalized by the predicted scaling N u−1 ,
versus the transition parameter, RaE 3/2 . The solid horizontal line indicates the predicted scaling
(4.1), and the dotted horizontal lines indicate the standard deviation of non-rotating convection
data about (4.1) b) The ratio of thermal boundary layer thickness to viscous boundary layer
thickness versus RaE 3/2 for rotating convection. The dashed vertical lines in both panels indicate
the approximate location of the boundary layer crossing. Symbols have the same meaning as in
figure 2.
L
L
a
b
L/2
3L/4
Pr = 1
Pr = 7
P r = 100
E=∞
Ra = 7 × 105
Height
Height
3L/4
L/4
0
0
L/2
E=∞
E = 10−3
E = 10−4
E = 10−5
Pr = 1
Nu ≈ 8
L/4
ΔT/4
ΔT/2
Temperature
3 ΔT/4
ΔT
0
0
ΔT/4
ΔT/2
Temperature
3 ΔT/4
ΔT
Figure 6. Mean temperature profiles for a) non-rotating and b) rotating convection. a) Temperature profiles for non-rotating convection with Ra = 7 × 105 and P r = 1, P r = 7, and
P r = 100. Thermal overshoot is observed as the Prandtl number is increased. b) Temperature
profiles for P r = 1 and: E = ∞ (non-rotating) and Ra = 7 × 105 , solid curve, as in panel a;
E = 10−3 and Ra = 7 × 105 , dashed curve; E = 10−4 and Ra = 4 × 106 , dash-dotted curve; and
E = 10−5 and Ra = 7 × 107 , dotted curve. All four cases have 7.5 < N u < 8.
5. Interior temperature gradients, β
The scaling relationship between heat flux and the thermal boundary layer thickness
(4.1) hinges upon the assumption that the interior fluid is well mixed. Figure 6 shows
mean temperature profiles for convection simulations with and without rotation. We observe that prevailing temperature gradients vary with P r for a given Ra in non-rotating
convection (panel a), and with E for a given P r and N u. In order to analyze these profiles
more systematically, we quantify the degree of thermal mixing in the bulk fluid by calculating the mean vertical temperature gradient at mid-depth, which is non-dimensionalized
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E. M. King, S. Stellmach, and B. Buffett
0.3
−0.001
a
b
0.25
−0.01
0.15
β
β
0.2
−0.1
0.1
0.05
−1
0
3
4
10
10
5
6
10
10
7
−2
8
10
−1
10
10
0
10
10
δ v /δ T
Rayleigh Number
−0.2
0
d
c
−0.4
β
β
−0.2
−0.5
−0.6
−0.8
−1
1
10
2
3
10
RaE
10
−1
7
10
4/ 3
15
25
RaE 4/ 3
Figure 7. Interior temperature gradients (5.1) for non-rotating (a) and rotating (b-d) convection. Symbols have the same meaning as in figure 2. a) Temperature gradients β, plotted against
Ra for non-rotating convection. b) Temperature gradients calculated from rotating convection
simulations plotted versus the ratio between Ekman and thermal boundary layers. The dashed
vertical line illustrates the boundary layer transition (see figure 5), and the dotted horizontal
line show the mean magnitude of β from the non-rotating simulations (panel a). c) Interior
temperature gradients calculated from rotating convection simulations plotted versus supercriticality, RaE 4/3 . The solid line shows β = 6.6RaE 4/3 . d) Shows an expanded view of the same
data as in panel c.
by the imposed, global gradient,
β≡h
∂T ′
(z = L/2)iH
∂z
(L/∆T ).
(5.1)
5.1. Non-rotating
Figure 7a shows calculations of the interior temperature gradients, β, plotted versus
the Rayliegh number for the non-rotating simulations. Two notable observations arise.
First, convection with high Rayleigh numbers (Ra > 106 ) and/or low Prandtl numbers
(P r = 1) are generally well mixed thermally. Second, the prevailing temperature gradients are positive, whereas the imposed global gradient is negative. This overshoot of
the background gradient can be seen explicitly in the thermal profiles plotted in figure 6a. Thermal overshoot has been observed previously for convection with low Re, and
is attributed to convective flows dominated by long-lived thermal plumes (e.g., Olson &
Corcos 1980). These plumes traverse the layer as a quasi-Stokes flow, and collect along
Page 11 of 21
Rotating convection scalings
11
the opposite boundary where their temperature anomalies produce local peaks in the
mean temperature profile.
5.2. Rotating
We observed in section 4.2 that rotation upsets the assumptions necessary to reach
(4.1) (see figure 5). Previous studies have shown that the Coriolis force is capable of
inhibiting turbulent mixing in general (Taylor 1921), in convection experiments (Boubnov
& Golitsyn 1990; Fernando et al. 1991) and simulations (Julien et al. 1996; Sprague
et al. 2006), and in convective dynamo simulations (King et al. 2010). Understanding the
interplay between rotation and thermal mixing is essential for adequate parameterization
of small scale convection in the oceans and core of Earth and other planets.
King et al. (2010) argue that the inhibition of thermal mixing for convection systems
occurs when the Ekman boundary layer is thinner than the thermal boundary layer, following arguments for N u scaling behavior (King et al. 2009). Our current results, shown
in figure 7b, are consistent with this suggestion. Figure 7b shows interior temperature
gradient calculations, β, are plotted versus the ratio between the measured thicknesses
of the boundary layers. The dashed vertical line indicates the boundary layer transition,
where δv = δT . The dotted horizontal line shows the mean magnitude of the mid-plane
gradient for the non-rotating simulations for comparison. We observe that rotation suppresses thermal mixing when the Ekman layer is the thinner boundary layer.
Figures 7c,d show the interior temperature gradient calculations plotted versus supercriticality, RaE 4/3 (onset of convection for the present range of E and P r should
occur near RaE 4/3 = 7 (Chandrasekhar 1953)). Sprague et al. (2006) observe that,
near onset for an asymptotically reduced system, temperature gradients scale with the
inverse of supercriticality. Our results are consistent with this observation. A best fit
power law scaling for the rotating convection simulations that have δE < 0.5δT yields
β = −6.2(±2.3)(RaE 4/3)−0.98(±0.14) . On average for these rapidly rotating simulations,
β/(RaE 4/3 ) = −6.6(±1.2), which is shown as the solid line in figures 7c,d, and fits these
data to within 6% on average.
6. Bulk length scales, ℓ
6.1. Non-rotating
It is often observed in experimental studies of Rayleigh-Bénard convection that thermal
turbulence organizes into large-scale circulation patterns (also known as mean thermal
winds or flywheel convection patterns) (Ahlers et al. 2009). Such aggregate patterns of
thermal turbulence result from the presence of rigid sidewalls, which help to organize
smaller scale turbulence into a sweeping cell that follows the edges of the container (Xi
et al. 2004). In the present simulations, however, the absence of rigid sidewalls eliminates
this selection process, and large scale circulation patterns are not observed. The length
scales for flow in our simulations are instead naturally selected, and should be the typical
separation of thermal plumes.
Figure 8 illustrates this plume separation, ℓ. We develop a scaling for ℓ by considering
the timescale for flow along the boundary. If fluid flows along the boundary with typical
speed ub , then the travel time between plumes is τ ∼ ℓ/ub . As this flow travels along the
boundary, a thermal front will propagate toward the interior. This diffusive growth is
illustrated in figure 8b as thermal profiles within the boundary layer flow. The boundary
flow should become unstable when this thermal anomaly becomes roughly as thick as
the mean thermal boundary layer. Thus, we can relate the boundary flow and plume
Page 12 of 21
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E. M. King, S. Stellmach, and B. Buffett
a
rp
rp
up
up
δT
ub
ub
b
δT
ub
T
Figure 8. A schematic of the plume separation length scale. The typical separation between
plumes is denoted by ℓ, the thermal boundary layer thickness by δT , typical plume speed by up ,
the typical plume width by rp , and the flow along the boundary by ub .
development timescale as:
τ ∼ ℓ/ub ∼ δT2 /κ.
(6.1)
up · πrp2 = ub · πℓδT .
(6.2)
Finally, we can determine how the boundary layer flow speed should scale with P e
in order to estimate the characteristic length scale ℓ. Figure 8a illustrates the threedimensional geometry of plumes and boundary layer. The plumes have radius rp and
travel vertically with speed up . Boundary layer flow spreads (converges) horizontally
along the boundary away from impinging plumes (toward departing plumes) with speed
ub . Conservation of mass flux dictates that
Assuming that up ∼ P e(κ/L), and that the plume width scales with thermal boundary
layer thickness, rp ∼ δT , we can combine (6.1) and (6.2) to estimate the natural plume
spacing in non-rotating convection:
3/2
δT
P e1/2 L.
(6.3)
ℓ∼
L
A scaling law for plume spacing in high P r convection was developed by Parmentier &
Sotin (2000) following similar reasoning.
Figure 9a shows calculations of the characteristic horizontal length scale ℓ versus the
Page 13 of 21
13
Rotating convection scalings
0.3
0.7
a
b
ℓ/ L
ℓ/ L
0.1
0.3
0.03
0.01
0.1
0.1
P e 1/ 2 (δ T /L ) 3/ 2
1
−6
10
−5
10
−4
10
−3
10
Ekman Number
Figure 9. Characteristic length scale for non-rotating (b) and rotating (b) convection. Symbols have the same meaning as in figure 2. a)ℓ/L versus P e1/2 (δT /L)3/2 (from (6.3)), from
non-rotating convection simulations. The solid line shows ℓ/L = 0.8P e1/2 (δT /L)3/2 . b) Typical
horizontal length scale versus E for rotating convection. The solid line shows ℓ = E 1/3 L.
this scaling (6.3) from non-rotating convection simulations. The scaling ℓ/L = 0.8P e1/2 (δT /L)3/2
is shown as a solid line, and describes the data to within 25% on average.
An alternative, but complementary estimate of interior length scales in non-rotating
convection can be determined by heat flux considerations in the bulk. If plumes typically
have radius rp and average spacing ℓ (see figure 8), then the fraction of a horizontal
cross-section through the bulk fluid occupied by plumes is π(rp /ℓ)2 . Mean convective
heat flux, N u − 1 = hu′ T ′ i, may scale with the typical plume speed up ∼ P e, multiplied
by this plume density fraction. If we again assume that the typical width of the plumes
scales with the thermal boundary layer thickness, rp ∼ δT , then
1/2
Pe
ℓ ∼ δT
.
(6.4)
Nu − 1
If we further assume that the interior fluid is well mixed such that δT ∼ 1/N u and
N u − 1 ≈ N u, then we recover the scaling derived by considering boundary layer flow
(6.3).
6.2. Rotating
Rotating RBC cannot be purely geostrophic, since mean convective heat transport requires vertical flow, which cannot be z-invariant in a layer of finite thickness. Convection
in the presence of strong Coriolis forces must break the Taylor-Proudman (TP) theorem.
For fluids with P r & 1, it is usually assumed that the TP theorem is broken by viscous
forces acting on small horizontal length scales. The characteristic length scale for kinetic
energy should then occur at the scale on which viscosity can balance the Coriolis term in
the horizontal direction. Taking the curl of the momentum equation and retaining only
these two terms, we have
∂u
∼ ν∇2 ω,
(6.5)
2Ω
∂z
where ω = ∇ × u is the fluid vorticity. The next-order terms in the TP theorem tells us
that ∂/∂z ∼ 1/L (e.g., Greenspan 1968; King & Aurnou 2012), so the lhs of 6.5 scales as
ΩU/L. The rhs of 6.5, in contrast, is dominated by horizontal gradients, ∇2 ω ∼ U/ℓ3 .
This interior balance therefore predicts vortical flow with characteristic horizontal length
Page 14 of 21
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E. M. King, S. Stellmach, and B. Buffett
scale
ℓ/L ∼ E 1/3 .
(6.6)
This characteristic scale also corresponds to the critical length scale for the onset of
convection in linear analysis of rapidly rotating convection (Chandrasekhar 1953).
Figure 9b shows calculations of typical horizontal length scale ℓ/L for rotating convection plotted versus the Ekman number. A best fit power law scaling yields:
ℓ/L = 1.09(±0.14) E 0.34(±0.01);
(6.7)
for the geostrophic cases (RaE 3/2 < 10), in agreement with the scaling prediction. The
solid line in figure 9b shows ℓ/L = E 1/3 , which fits the rotating convection data to
within an average of 3%, and fits within 1% on average for data in the geostrophic
regime (RaE 3/2 < 10).
7. Flow speeds, P e
7.1. Non-rotating
Much work has been done to relate the typical speed of convective motions (P e) with
the strength of driving (Ra) in both laboratory experiments and numerical simulations
(e.g., Ahlers et al. 2009). A useful tool for such scaling is the exact balance between mean
global production and dissipation of kinetic energy:
κ2
(7.1)
(N u − 1)Ra = h(∇u)2 i,
h4
which follows from taking the global average of u·(1.1) (e.g., Shraiman & Siggia 1990).
The rightmost term can be approximated as either U 2 /ℓ2 or U 2 /δv2 , by assuming that the
relevant length scale for dissipation is either the characteristic interior length scale, ℓ, or
the boundary layer thickness, δv , respectively (Grossmann & Lohse 2000). Unfortunately,
in our simulations we find no regimes in which dissipation is dominated by either the
bulk or the boundary layers, in agreement with the results of Calzavarini et al. (2005).
Substitution of each of these approximations into (7.1) produces scalings for convective
speed based on dissipation in the bulk,
P e∗int ≡ (N u − 1)1/2 Ra1/2 (ℓ/L),
(7.2)
and dissipation in the boundary layers,
P e∗bl ≡ (N u − 1)1/2 Ra1/2 (δv /L),
(7.3)
Figures 10a and 10b show Peclet number calculations plotted versus these dissipation
∗ 1.04(±0.04)
integral scalings. A best fit power law regression to each gives P e = 0.28(±0.06) P eint
∗ 0.99(±0.03)
for the bulk dissipation estimate, and P e = 0.59(±0.09) P ebl
for the boundary
layer dissipation estimate. Imposing linear fits, we get P e/P e∗int = 0.36(±0.05) and
P e/P e∗bl = 0.56(±0.07), which are shown in figures 10a and 10b, respectively. An f-test
reveals that these fits have statistically indistinguishable differences.
These scalings suggest that convective flow may strike a balance between dissipation
in the bulk and boundary layers. Such a balance can also be seen in the time-averaged
heat equation,
∂2
∂
uT ′ = κ 2 T ,
(7.4)
∂z
∂z
where T ′ is a typical temperature fluctuation. Assuming an isothermal interior and welldefined thermal boundary layers, we can scale this balance as U ∆T /L ≈ κ∆T /δT2 . Using
Page 15 of 21
15
Rotating convection scalings
4
4
10
10
a non-rotating
b non-rotating
Peclet Number
Peclet Number
3
10
3
10
2
10
2
10
1
10
1
10
0
0
10
1
2
10
10
3
10
4
10
1
10
2
10
10
P e ∗int
4
4
10
4
10
10
c non-rotating
d rotating
3
Peclet Number
Peclet Number
3
10
P e ∗bl
10
3
10
2
10
2
10
1
10
1
10
0
10
1
3
10
30
Nusselt Number
1
10
2
10
3
10
4
10
P e ∗int
Figure 10. Peclet number scalings for non-rotating (a-c) and rotaing (d) convection. Symbols have the same meaning
p as in figure 2. a) Peclet number versus the interior dissipation scaling, P e∗int =
Ra(N u − 1)(ℓ/L), for non-rotating convection. The solid line
shows the scaling P ep= 0.36P e∗int . b) Peclet number versus the boundary layer dissipation scaling, P e∗bl =
Ra(N u − 1)(δv /L), for non-rotating convection. The solid line shows
P e = 0.56P e∗bl . c) Peclet number versus the Nusselt number for non-rotating convection.
2
The solid
p line shows P e = 2.5N u . d) Peclet number versus the interior dissipation scaling,
∗
P ebl = Ra(N u − 1)(δv /L), for rotating convection. The solid line shows P e = 0.65P e∗int .
(4.1), this balance gives a Peclet number scaling in terms of the Nusselt number alone,
P e ≈ 4N u2 .
(7.5)
Figure 10c shows P e plotted versus N u. A best fit power law regression yields P e =
2.0(±0.3) N u2.11(±0.07). When N u > 4, for which (4.1) was shown to hold, our data
agree better with (7.5), as P e = 3.1(±0.8) N u1.93(±0.11). Imposing a scaling of the form
(7.5) gives P e/N u2 = 2.5(±0.4), which gives an average misfit of 21% for all non-rotating
data and 11% for N u > 4, and is shown as the solid line in figure 10c. We caution, though
that, since this scaling is derived solely from the heat equation, it is unlikely to hold for
very large Reynolds numbers, when inertia is expected to play a dominant role. And we
do observe that the quality of the fit declines for decreasing P r and increasing Ra.
7.2. Rotating
As the rate of rotation increases (decreasing E), we find that viscous dissipation is increasingly dominated by its interior (bulk) contribution. We therefore expect that flow
Page 16 of 21
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E. M. King, S. Stellmach, and B. Buffett
speeds will follow the interior dissipation scaling (7.2). Figure 10d shows calculations of
convective flow speed P e plotted versus the interior dissipation scaling (7.2). A best fit
∗ 0.99(±0.04)
for all rotating convection
power law regression yields P e = 0.62(±0.14) P eint
simulations, in agreement with the prediction (7.6). The average value of P e/P e∗int is
0.65, which is shown as the solid line in figure 10d, and fits all of the rotating convection
data with an average error of 36%.
The Coriolis force can do no work, and so only enters the energetic flow speed scaling
(7.2) through its influence on length scales. Fortunately, the characteristic length scales
of rapidly rotating convection follow well known scaling relationships in terms of E alone
(6.6). Substituting (6.6) into (7.2) produces a prediction for flow speed based on the
input parameter E rather than calculations of length scales:
P e ∼ P e∗V AC ≡ (N u − 1)1/2 Ra1/2 E 1/3 .
(7.6)
Here, the subscript ‘V AC’ stands for visco-Archimedean-Coriolis, for the triple force balance on which it is based. This scaling provides a similar quality-of-fit to the scaling
using actual, calculated lengtch scales, and so is not plotted here. Best fit power law re∗ 0.94(±0.03)
for all rotating convection simulations, and
gressions yield P e = 0.9(±0.2) P eV AC
∗ 1.01(±0.01)
P e = 0.66(±0.04) P eV AC
for geostrophic simulations (RaE 3/2 < 10), in agreement with the prediction (7.6). The average value of P e/P e∗V AC for RaE 3/2 < 10 is
0.75, and P e = 0.75P e∗V AC fits all of the rotating convection data with an average error
of 43%, and fits the geostrophic cases (RaE 3/2 < 10) to within 8% on average.
8. Discussion
Many of the scaling laws presented here can be combined to produce new scalings
without reducing agreement with data. For example, combining (7.5) with (6.3) and
(4.1) generates a scaling for the characteristic length scale in non-rotating as a function
of the thermal boundary layer thickness alone,
ℓ/L ∼ (δT /L)1/2 .
(8.1)
This scaling describes the data as well as (6.3): with a prefactor of 0.73, it fits the nonrotating data to within 16% on average (and quality-of-fit improves with increasing Ra).
Furthermore, this scaling is identical to that observed by Parmentier & Sotin (2000) and
Zhong (2005).
Similarly, for the non-rotating viscous boundary layer thickness, we can combine (3.3),
(7.5), and the classical heat transfer scaling, N u ∼ Ra1/3 , to produce:
δv /L ∼ Ra−1/6 P r1/4 .
(8.2)
We emphasize that this scaling is empirically based, since we offer no theoretical explanation for (3.3). This relationship is, however, in close agreement with the empirical
scaling observed in Lam et al. (2002), whose non-rotating experiments span the ranges
106 . Ra . 1011 and 10−2 . P r . 103 , as well as the numerical work of Breuer et al.
(2004), who utilize a similar geometry to ours, but reach Re ≈ 2 × 104. We feel this agreement substantiates our finding that δv ∼ Re−1/4 rather than the typical Blasius scaling,
δv ∼ Re−1/2 . This difference is important as many theoretical treatments of the turbulent convection problem assume the latter (e.g., Grossmann & Lohse 2000). We suggest
that the Blasius scaling is not observed in these simulations and experiments because the
boundary layer is active, rather than a passive response to bulk turbulence assumed in
(3.1) and (3.2). Qiu and Xia measure viscous boundary layer thicknesses along both the
Page 17 of 21
17
Rotating convection scalings
sidewall (1998a) and bottom (1998b) boundaries. They find that the sidewall boundary
layer has Blasius-type Re dependence, while the bottom boundary layer thickness has
a weaker dependence, scaling closer to (8.2). (For Boussinesq convection, we expect the
same holds for the top boundary.) This supports the argument that the top and bottom boundary layers, through which heat is fluxed, are dynamically active and so have
thicknesses that scale differently from the Blasius type. At the sidewall, however, which
is roughly adiabatic, one might expect that the boundary layer is a passive response
to bulk turbulence. The Blasius boundary layer scaling is also recovered in experiments
where a strong mean wind is imposed by the tank geometry (Sun et al. 2008).
Our simulations use periodic sidewall boundary conditions in order to eliminate the
effects of sidewalls, which are absent in the original formulation of the Bénard problem.
The results of the non-rotating convection simulations shown here suggest that the important length scales in RBC, δT , δv , and ℓ, are dynamically interdependent. Combining
(3.3), (7.5), and (4.1), we get another scaling relationship for the viscous boundary layer
in non-rotating convection,
δv /L ∼ (δT /L)1/2 P r1/4 .
(8.3)
Scaling relationships (8.1) and (8.3) describe the connections between these length scales.
Rotating convection is somewhat simpler in this regard, as the scaling behavior of δv
and ℓ are well-behaved responses to particular balances between the Coriolis force and
viscosity. The opposite is true for δT : in non-rotating convection the interior is effectively
mixed such that (4.1) roughly holds. In rotating convection, the Coriolis force inhibits
this mixing, permitting finite interior temperature gradients (figure 7), and complicating
the usual simple thermal boundary layer controlled depiction of heat transfer.
Of course, care must be taken when amalgamating or extrapolating the scalings presented here. For example, for non-rotating flow speeds, we can combine (4.1), (7.2), and
(8.1) to get, in the limit of high N u,
P e ∼ Ra1/2 .
If, however, we instead use (7.5) and (4.1) with N u ∼ Ra
P e ∼ Ra2/3 .
(8.4)
1/3
, we arrive at
(8.5)
We emphasize in section 7.1 that (7.5), and therefore (8.5), is likely to hold only for
low Re (high P r and/or low Ra). We expect (8.4) should, in contrast, hold for high
Re. Experiments (Lam et al. 2002) and simulations (Silano et al. 2010) exploring broad
ranges of P r support this expectation.
If we maintain the N u dependence in the P e scaling for non-rotating convection, we
find better convergence. Using (4.1) with (7.2) and (8.1) (or, equivalently, with (7.3) and
(8.3) and neglecting the P r-dependence in the latter), we arrive at
P e ∼ (Ra − Ra/N u)1/2 .
(8.6)
This scaling law is shown in figure 11a with a prefactor of 0.18, which fits the non-rotating
convection data to within 14% on average.
Previous work has investigated transitions in heat transfer behavior for rotating convection by comparing N u from rotating and non-rotating experiments and simulations.
We can do the same for flow speeds, P e. First, we must characterize the non-rotating P e
behavior in a simple way. For this, we take a best fit power law regression of P e with Ra
for E = ∞ and Ra > 105 , which yields P e = 0.14(±0.1)Ra0.52(±0.04). Figure 11b shows
flow speeds from rotating convection simulations normalized by this scaling, P eRa−0.52 ,
plotted versus the transition parameter, RaE 3/2 . We observe a change in behavior near
Page 18 of 21
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E. M. King, S. Stellmach, and B. Buffett
0
4
10
10
Peclet Number
a
b
3
10
P eRa −0.52
−1
2
10
1
10
−2
10
10
0
10 2
10
−3
3
10
4
10
5
10
6
10
Ra − Ra/Nu
7
10
8
10
10
0
10
1
10
2
RaE 3 / 2
10
3
10
Figure 11. a) Peclet number plotted versus Ra−Ra/N u from (8.6) for non-rotating convection.
The solid line depicts P e = 0.2(Ra−Ra/N u)1/2 . b)Transition between rotating and non-rotating
flow speed behaviors. Peclet numbers from rotating convection simulations are normalized by the
non-rotating fit Ra0.52 and plotted agains the boundary layer transition parameter. The solid
horizontal line shows the non-rotating fit, P e = 0.14Ra0.52 . The dotted vertical line indicates
the boundary layer transition from figure 5b.
the boundary layer transition RaE 3/2 ≈ 10 (see figure 5b) between suppressed flow
speeds and flow speeds that conform to the non-rotating behavior indicated by the horizontal line. In agreement with the heat transfer transitions observed in King et al. (2009,
2012), this
p flow speed transition is not well described by the convective Rossby number,
Roc = RaE 2 /P r. Thus, our simulations indicate that the scaling behavior of both
global heat transfer and mean flow speeds are governed by the relative thicknesses of the
thermal and Ekman boundary layers.
One perhaps surprising result of note is that our quantification of the influence of rotation does not agree with that by the oft-used Rossby number. Not to be confused with
the convective Rossby number, the Rossby number, Ro = P eE/P r, is used to characterize the relative importance of inertial and Coriolis forces. It is often assumed that fluid
systems with low Rossby number (Ro < 1) should be geostrophic. We find instead that
the the influence of rotation on the quantities of interest here is better described by the
transition parameter RaE 3/2 , which characterizes the relative thicknesses of the thermal
and Ekman boundary layers (figure 5b). In our simulations, we observe weakly rotating
convection cases (RaE 3/2 > 10) that have Rossby numbers as small as Ro . 10−3 . Furthermore, cases with Ro < 1 are found even at our highest finite value of RaE 3/2 = 7000.
Our results therefore indicate that low Rossby numbers do not necessitate geostrophic
convection.
9. Summary
We investigate the scaling behavior of five quantities produced in rotating and nonrotating convection simulations in a cartesian box with periodic sidewalls, operating
within the parameter ranges 10−6 ≤ E ≤ ∞, 1 ≤ P r ≤ 100, and 103 . Ra . 109 . The
quantities of interest are the viscous boundary layer thickness, δv , thermal boundary
layer thickness, δT , mid-layer mean temperature gradient, β, characteristic bulk length
scale, ℓ, and mean flow speeds, P e. Table 3 summarizes our results.
We find three important convection regimes: geostrophic convection, RaE 3/2 . 10,
Page 19 of 21
Rotating convection scalings
19
Table 3. Summary of results. Regarding the weakly and non-rotating heat transfer scalings, we
stress that the 2/7 law is empirical, and likely increases with Ra to a 1/3 law (e.g., Niemela &
Sreenivasan 2006; Niemela et al. 2010).
Page 20 of 21
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E. M. King, S. Stellmach, and B. Buffett
where flow speeds and heat transfer are suppressed by the Coriolis force; weakly rotating
convection, 10 . RaE 3/2 < ∞, where flow speeds and heat transfer are not strongly
affected by rotation, but δv and ℓ are still dictated by the Coriolis force; and non-rotating
convection. The transition from geostrophic to weakly rotating convection occurs when
the thermal boundary layer becomes thinner than the Ekman layer.
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