Finite Prandtl Number 2-D Convection at High
Rayleigh Number
Catherine Hier Majumder1,2, David A. Yuen1,2, Erik O. Sevre1,2, John M. Boggs2,
and Stephen Y. Bergeron2
1. Department of Geology and Geophysics, University of Minnesota, Minneapolis, MN
2. Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN
submitted to Electronic Geosciences
http://www.msi.umn.edu/~cathy/prpaper/paper.html
Abstract
Finite Prandtl number thermal convection is important to the dynamics of planetary
bodies in the solar system. For example, the complex geology on the surface of the
Jovian moon Europa is caused by a convecting, brine-rich global ocean that deforms the
overlying icy "lithosphere". We have conducted a systematic study on the variations of
the convection style, as Prandtl numbers are varied from 7 to 100 at Rayleigh numbers
106 and 108. Numerical simulations show that changes in the Prandtl number could exert
significant effects on the shear flow, the number of convection cells, the degree of
layering in the convection, and the number and size of the plumes in the convecting
fluid. We found that for a given Rayleigh number, the convection style can change from
single cell to layered convection, for increasing Prandtl number from 7 to 100. These
results are important for determining the surface deformation on the Jovian moon
Europa. They also have important implications for surface heat flow on Europa, or for
the interior heat transfer of the early Earth during its magma ocean phase.
Introduction
The Prandtl number is the ratio of the kinematic viscosity to the thermal diffusivity of a
fluid and is an important control variable in thermal convection. Although infinite
Prandtl number approximations are appropriate for the Earth’s crust and mantle, finite
Prandtl number fluids are also ubiquitous throughout our Earth. They include many of
the fluids associated with our everyday lives, such as air, water, alcohols, and oils.
Convection in air, which has a Prandtl number of about 1, is responsible for the weather.
Convection in water, with a Prandtl number of 7, is important for oceanic currents. The
liquid iron of the Earth’s outer core is a finite Prandtl fluid. Convection in this
electrically conducting fluid generates the Earth’s magnetic field (Glatzmaier and
Roberts, 1995) . Finite Prandtl convection is important to crystallization processes in
magma chambers (Bergantz, 1992). Convection in a magma ocean under high Rayleigh
and Prandtl number conditions was likely to have occurred early in Earth’s history.
A knowledge of convection styles in this type of fluid may give important insights into
the heat flux and chemical differentiation of the early Earth. Finite Prandtl number flows
are especially important to the dynamics of the outer planets and their moons. The
surface of Jupiter is formed by convection in a fluid with a Prandtl number of around
0.01 (Zhang and Schubert, 2000). The Galileo missions have confirmed the existence of
convecting oceans on the Jovian moons Europa and Callisto (Carr, 1998, Khurana,
1998). Thomas and Delaney (2001) have used numerical models to show that a sea-floor
spreading and hydrothermal vent system similar to the Earth’s mid-ocean ridges could
produce convection in Europa’s global ocean. A magnetic field is induced as the salty,
convecting, global oceans on the moons Europa and Callisto are dragged through
Jupiter’s magnetosphere (Khurana, 1998). Europa’s icy lithosphere has a wealth of
surface geology created by convection in its global ocean (Carr et al., 1998). Evidence
from the Voyager missions indicates that the surface features of Ganymede may be due to
convection in a water-ice mantle, which is overlain by an icy lithosphere (Kirk, 1987). A
better understanding of finite Prandtl number convection at high Rayleigh numbers will
allow us to better model heat flux and surface deformation on the Jovian moons.
Numerical simulations of Rayleigh-Bénard convection at both high Rayleigh and Prandtl
number are difficult. As the Rayleigh number increases, strong inertial forces generate
complex structures in the temperature and velocity fields that require increased grid
points and smaller timesteps. Increasing the Prandtl number leads to sharp temperature
gradients and strong shear flows. These effects also require increasing grid points and
decreasing timesteps. The problem is compounded by the fact that as the Prandtl number
is increased the fluid becomes stiffer and more time is required for movement to develop
from an initial temperature and velocity perturbation. Vincent and Yuen (1999, 2000)
have conducted numerical simulations for a Prandtl number of unity (air) at Rayleigh
numbers from 108 to 1014. Numerical simulations have been conducted for Prandtl
number 7 (water) up to Rayleigh numbers of 1013 (Yuen et al., 2000). These simulations
showed the complex structures generated by finite Prandtl convection, and wavelets to be
an ideal tool for understanding the multiscale nature of these structures (Yuen et al.,
2000).
Heat flux scaling laws tend to show that the Nusselt number is proportional to (RaPr)n
where n is 1/3 for plume theories with a single length scale and 2/7 for plume theories
with several length scales (Zaleski, 2000). Grossmann and Lohse (2000) suggest that a
single power law may not appropriately explain the dependence of the Nusselt number on
the Rayleigh and Prandtl numbers since more than one heat transport regime may be
operating in a given experiment. Instead they propose that linear combinations of several
power laws may be necessary. They also indicate that for Pr > 7 formation of the wind
of turbulence, on which classic scaling theories are based becomes less likely. The wind
of turbulence is associated with shearing in the boundary layers that moves plumes in the
horizontal direction and results in a bulk stirring of the fluid. Verzicco and Camussi
(1999) conducted numerical experiments for Prandtl numbers ranging from 0.022
(mercury) to 15 at Rayleigh number 6 x 105. They found that below Pr = 0.35 heat is
transferred by large-scale recirculation cells; above Pr = 0.35 the heat is transferred by
thermal plumes. The aspect-ratio (A) may also play an important role in the heat
transport, as suggested by the Shraiman and Siggia (1990) theory, in which the Nusselt
number is predicted to be proportional to Ra2/7Pr-1/7 A-3/7.
The Prandtl and Rayleigh numbers of some convecting systems are shown in Fig. 1 and
Table 1. Previously, fluids with Prandtl numbers as different as the Earth’s mantle with
Prandtl number 1025 and mushy ice with Prandtl number 104, such as found in Europa’s
lithosphere, are both approximated as infinite Prandtl number fluids. Significant
differences may exist in the convection styles of fluids due to changes in Prandtl number
in the intermediate range from Pr = 1 to 100.
Fig. 1. Rayleigh and Prandtl
numbers of some convective
systems.
System
Rayleigh Prandtl
Number Number
Pot of
1x107
Water
Europa’s
1x1025
Ocean
Europa’s
ice
1x1018
lithosphere
Earth’s
5x106
Mantle
7
7
1x104
1x1024
Table. 1. Rayleigh and Prandtl numbers of some convective systems.
This study focuses on fluids with Prandtl numbers ranging from 7 to 100. Fluids in this
range of Prandtl numbers have important industrial applications. For example, ethanol
has a Prandtl number of 17. A solution of 60% glycerin and 40% water has a Prandtl
number of 100. Shevtsova et al. studied the onset of thermocapillary convection in liquid
bridges for fluids with Prandtl numbers up to 35. This type of convection can cause
undesired compositional variations in crystals grown industrially.
We simulated convection at Rayleigh numbers of 106 and 108. The Prandtl numbers in
this study are still significantly below those of some important geologic fluids such as the
mushy ice that forms diapirs on Europa, or basaltic magma. Both of these fluids have
Prandtl numbers of order 104. This study represents a first step in the process of
simulating systems such as Europa. It allows us to begin to asses the importance of
Prandtl number in the behavior of Rayleigh-Bénard convection at higher Prandtl
numbers. It represents the beginning of the development of techniques that will allow us
to conduct simulations at even higher Prandtl and Rayleigh numbers in the future.
We have conducted simulations in an aspect-ratio 3 box for Prandtl numbers of 7, 17, 40,
80, and 100 at a Rayleigh number of 106 and for Prandtl numbers of 7, 17, 40, 60, 80, and
100 at a Rayleigh number of 108. The convection simulations have a complex time
evolution, which is discussed in detail for the simulations of Prandtl number 7 and
Rayleigh number 106. We have illustrated the time evolution through the use of
animations. We have conducted wavelet analysis on the long-term solutions of the
temperature deviation and vorticity fields for each of the simulations to help characterize
the convection style.
Wavelets have been used as a tool to better understand the convection patterns captured
by seismic tomography of the Earth’s mantle (Bergeron et al., 1999, 2000, Yuen et al.,
2000). One dimensional wavelet transforms have been used for many years in the
analysis of time series, such as the interseasonal Earth rotation variations (Chao and
Naito, 1995). Malamud and Turcotte (2001) analyzed one-dimensional tracks of Martian
topography using wavelets. Vecsey and Matyska (2001) used wavelets to analysis
temperature, kinetic energy, and Nusselt number time series in mantle convection
simulations. Castillo et al. (2001) used wavelet transforms of seismograms to
characterize the transition zone structure below California.
Wavelets are also a powerful tool for processing 2-D convection simulation data, such as
temperature and vorticity fields (Hier et al., 2000). Temperature and vorticity maps from
convection simulations can be quite complicated and difficult to interpret at higher
Rayleigh and Prandtl numbers. The wavelet transform allows us to separate the
large-scale features, such as convection cells, from smaller scale features, such as shear
layers. This leads to a better understanding of the different processes that produce the
temperature and vorticity maps.
In order to take advantage of the web format for paper publishing, we have used figures
extensively in this paper. We focus on the temperature deviation and vorticity fields
generated by the simulations along with their wavelet transforms. The finite Prandtl
simulations resulted in rich patterns in the temperature deviation and vorticity fields that
is most easily conveyed by figures. The time evolution of the simulations was also very
rich. The web format has allowed us to feature in this communication animations of how
the complex dynamics develop with time.
Numerical Model
The nondimensional equations of finite Prandtl number convection in the Boussinesq
approximation are:
(1)
(2)
(3)
where t is the time nondimensionalized by the thermal diffusivity, T is the temperature
deviation from the conductive profile, V is the velocity vector, P is the pressure, and ez is
the unit vector from the top to the bottom of the fluid layer. Ra is the Rayleigh number
for Rayleigh-Bénard convection in a basely heated box, and Pr is the Prandtl number.
The numerical method used is a spectral method for 2-D convection with finite Prandtl
number (Vincent and Yuen, 1999) with sine and cosine series expansion as the basis
functions. The top and bottom of the box have free-slip boundary conditions. The
time-marching uses a mixed leap frog-Cranck-Nicholson two-step pressure correction
scheme. The grid size for each simulation is shown in Table 2. The aspect-ratio of the
box is defined as the width of the box over the layer depth.
Rayleigh Number
Prandtl Number
Aspect-Ratio
(width x depth)
Grid Size
(width x depth)
106
7
3x1
768x256 points
106
17
3x1
768x256 points
106
40
3x1
768x256 points
106
80
3x1
768x256 points
106
100
3x1
1080x360 points
108
7
3x1
1080x360 points
108
17
3x1
1080x360 points
108
40
3x1
1080x360 points
108
60
3x1
1080x360 points
108
80
3x1
1080x360 points
108
100
3x1
1080x360 points
Table 2. Grid size for simulations.
Wavelet Transform
Wavelet transform (e.g. Resnikoff and Weiss, 1998) allows one to study the scales within
a dataset without losing the spatial information. Whereas the Fourier transform or
spherical harmonics provides a global description of a wavelength over a dataset; the
wavelet transform provides a localized description of a particular length scale at each grid
point in the multidimensional dataset (van den Berg, 1999). The 2-D wavelet transform
is:
(4)
where vector b is the two-dimensional position parameter, scalar a is the scaling
parameter, and Lx and Ly are the lengths of the periodic rectangle in Cartesian space
(Bergeron et al., 2000). We have used as the mother wavelet the Mexican Hat, or the
second-derivative of the Gaussian function (Daubechies, 1992):
(5)
The wavelet transform is done in Fourier space to simplify and speed-up the calculation
(Bergeron et al., 2000).
Scalogram
The result of a Fourier transform is a coefficient describing the strength of a given
wavelength over the whole dataset. The spatial or phase information is not explicitly
preserved. Therefore, you cannot compare the strength of a given wavelength at two
different points.
The wavelet transform produces a coefficient describing the strength of a given length
scale at each data point. The wavelet transform of a 2-D dataset is a 3-D function known
as a scalogram. The scalogram is a function of both the position vector and the scale.
The spatial information is explicitly preserved in the scalogram. This allows one to
analyze how the length scale varies with position. One can say that large length scales
are more important at point a, and small length scales are more important at point b.
Since a scalogram is a function of both scale and position, the dimension of the original
dataset is increased by one. Scalograms of our two-dimensional vorticity fields are
three-dimensional boxes. The extra dimensionality makes the scalograms difficult to
store and visualize because of the large memory required. The 3-D boxes contain vast
amounts of data that cannot be easily digested by the eye when displayed on a computer
screen. Therefore, we took slices of the box at a given scale. For each scalogram we took
3 slices: one at a large-scale, one at a medium-scale, and one at a small-scale. These
individual slices are a function of position only. They show the importance of features at
a given scale at different positions in space (Fig. 12).
E-max and k-max
The scale parameter in the wavelet transform (Eq. 4) results in a dimensionality of one
degree higher than the physical space analyzed. This can make the scalograms difficult
to visualize and interpret. Analysis of E-max and k-max distributions is used to
synthesize the scalogram (Bergeron et al., 2000, Yuen et al., 2000). E-max represents the
energy of the scale that has the largest energy value at a given position. An E-max map
plots the energy value of the scale with the highest energy at a given point in space. It,
therefore, selects the energy value from the scale which is most important at that point.
We define the energy of the wavelet transform as the L2-norm:
(6)
Since the sign of both the thermal anomaly and the vorticity is important in convection
studies, we have multiplied the L2-norm by the original sign of the thermal anomaly or
vorticity function.
The k-max is the wavenumber at which maximum energy occurs for a given position
vector b. A k-max map plots the scale that is most important at a given point. The scale,
a, is related to the wavenumber, k, by:
(7)
where ß is a tuning parameter, which we have chosen as 0.22. High k-max values
represent small-scale features. These features are associated with areas of strong
gradients. Therefore, the k-max map is an excellent tool for picking out areas of strong
gradients in a dataset. For temperature deviation and vorticity fields from convection
simulations, the areas of sharp gradients are associated with fluid movement. We will
use the k-max maps to emphasize areas of relative movement within the fluid.
The E-max and k-max analysis results in a lower dimensional approximation of the data.
It allows us to look at the data over a given range of scales. For example, to study
small-scale features in the dataset we would find E-max and k-max maps for scales
varying from k = 15 to k = 20.
Temporal Evolution
The temporal evolution of the simulations is best illustrated by an animation. Animation
1 shows the temporal evolution for the Ra = 106 and Pr = 7 simulation. An example of
a fluid with Pr = 7 is water at 20oC and 1 atm (Weast, 1987). The animation shows the
temperature deviation, which is the total temperature minus the conductive profile, and
the vorticity.
Animation 1. Temporal Evolution of Ra = 106 and Pr = 7
simulation for an aspect-ratio of 3.
Click on picture below to start animation.
The simulations are begun from an initial temperature deviation and vorticity
perturbations in the form of a sine wave with fundamental wavelength Lx/2 (Fig. 2). The
number of gridpoints for the simulation is specified in Table 2.
Fig. 2. The dimensionless time, t =
0.000 for Ra = 106, Pr = 7.
For Ra = 106 and Pr = 7, plumes and circulation cells develop from these perturbations.
The plumes have thin stems with thicker plumeheads that spread horizontally along the
thermal boundary layers (Fig. 3). The vorticity shows that the circulation cells have
pronounced shear layers along the edges with relatively weak rotation in their cores (Fig.
3). This indicates that the majority of the movement is along the edge of the convection
cells. Hot material rises vertically in plumes along on edge of a convection cell, and cold
material descends vertically in downwellings along the opposite edge. There is only
weak movement in the central area of the circulation cell.
Fig. 3. The dimensionless time, t =
0.405 for Ra = 106, Pr = 7.
As the Ra = 106, Pr = 7 simulation advances to t = 0.410, the plumeheads spread
horizontally across the thermal boundary layer and begin to impinge on neighboring
plumeheads, they move down forming regions of hot material below the upper thermal
boundary layer (Fig. 4). This creates a new thermal boundary layer. New plumes begin
to develop from the original stem region below the hot regions. In the vorticity field the
center of the circulation cells reverse flow directions (Fig. 4). The sharp shear layers
along the edges of the circulation cells remain in the same orientation as in Fig. 3. These
shear layers are associated with the new plumes arising from the original stem area.
Fig. 4. The dimensionless time, t =
0.410 for Ra = 106, Pr=7.
In the next frame of the temperature deviation we can see new plumes and downwellings
originating from the new central thermal boundary layers (Fig. 5). The vertical
symmetry, which originates from the symmetrical initial condition, is still maintained.
The plumes and downwellings growing from the original lower thermal boundary layer
begin to form caps below the lower thermal boundary layer. The original vorticity cells
begin to divide, thus forming two separate layers of circulation cells. This results in
layered convection (Fig. 5).
Fig. 5. The dimensionless time, t =
0.415 for Ra = 106, Pr = 7.
At time t = 0.420, the plumes growing from the center become connected with those
growing from the bottom of the box to form a branched plume structure (Fig. 6). New
plumes develop at the junction of this branch.
Fig. 6. The dimensionless time, t =
0.420 for Ra = 106, Pr = 7.
At t = 0.430, the layering structure still persists. The plume branching results in a
vorticity pattern that is a combination of a whole box circulation with a layered structure
(Fig. 7). We can still see the original four circulation cells that cover the whole height of
the box. These four cells are now each divided into three separate layers with the
strongest circulation in the middle layer.
Fig. 7. The dimensionless time, t =
0.430 for Ra = 106, Pr = 7.
At t = 0.473, the layered circulation cells divide to form vortex pairs (Fig. 8). Small
plumes begin to rise from both the lower and middle thermal boundary layers, and
downwellings form at both the upper and middle thermal boundary layers. The new
plumes and downwellings are evenly spaced along the length of the boundary layers.
The symmetry originating from the initial condition is preserved.
Fig. 8. The dimensionless time, t =
0.473 for Ra = 106, Pr = 7.
As convection progresses to t = 0.490, however, the layering that has developed
disappears (Fig. 9). As can be seen in the vorticity field, the four original circulation
cells are no longer divided horizontally (Fig. 9). Instead they are each divided vertically
into three long narrow cells. At t = 0.490, the symmetry which had been carried along
from the initial condition is lost.
Fig. 9. The dimensionless time, t =
0.490 for Ra = 106, Pr = 7.
At t = 0.525, the layered mode is destroyed, and the circulation cells take up the whole
box height (Fig. 10). Although the four circulation cells are still visible, the circulation is
reversed from that of the original cells. There is a strong horizontal shear component
resulting in a wind that sweeps across the box. The temperature deviation shows that the
plumes move up from the lower thermal boundary layer along thin stems, which are tilted
by the horizontal shear (Fig. 10). The plumes and downwellings spread out horizontally
once they reach the top and bottom with relatively thick thermal boundary layers.
Fig. 10. The dimensionless time, t =
0.525 for Ra = 106, Pr = 7.
The long-term solution is reached at t = 0.534. In the long-term solution there are still
four convection cells, but two of the cells have become wider at the expense of two cells
which have narrowed (Fig. 11). The horizontal shear remains in the long-term solution.
This results in shear layers on the edge of the wide convection cells.
Fig. 11. Long-term solution, at the
dimensionless time, t = 0.534 for Ra = 106, Pr = 7.
Wavelet Analysis
The use of wavelets allows us to separate the structures seen in the temperature deviation
and vorticity datasets by the scale of the features. This helps us to interpret the processes
operating at different scales in the convection pattern. The long-term solution of the
temperature deviation and vorticity fields for each simulations was transformed using the
continuous wavelet transform (Eq. 4) with the Mexican Hat as the mother wavelet (Eq.
5). The scalogram was computed from local wavenumber k = 1 to k = 20, which
corresponds to a resolution varying from 65 to 1.2% of the horizontal box width. The
resulting scalogram for Ra = 106 and Pr = 7 is a 16 MB file, which consists of a 3-D
box with 768 x 256 x 20 points. The scalogram is difficult to visualize due to the large
memory required and to the fact that it contains more information then can be easily
digested by the eye. Therefore, for better interpretation and to minimize data storage, we
have taken slices of the scalogram at 3 scales: k = 5, k =11, and k = 20, which
corresponds to resolutions of 27, 7.3, and 1.2% of the box width (Fig. 12 & 13).
For the simulation Ra = 106 and Pr = 7 the wavelet transform was performed on the
temperature deviation and vorticity fields shown in Fig. 11. The resulting scalogram
slices are shown in Figs. 12 & 13. In the large-scale slice of the temperature deviation
(Fig. 12), the only visible structure is the partitioning of the field into a cold bottom layer
and a top hot layer. In the medium-scale we can clearly discern the shapes of the
individual plume heads expanding horizontally across the boundary layers. The
small-scale structures are dominated by the plume tails. For the hot plumes, the plume
tails in the small-scale are surrounded by cold signatures. The opposite occurs for the
cold plumes. These false signals surrounding the plume are due to the attempt to fit the
structure to the edges of the Mexican Hat wavelet. The true sign of the signal is
associated with the central color.
Fig. 12. Scalogram of the Ra = 106 and
Pr = 7, associated with the long-term solution temperature deviation field in Fig. 11.
For the large-scale of the vorticity scalogram (Fig. 13), the wavelets pick out the two
wide and two narrow convection cells. In the medium-scale we can see the the four
convection cells along with stronger areas of circulation created on their edges by the
horizontal shear. The small-scale picks up the strong shear layer associated with the
edges of the convection cells.
Fig. 13. Scalogram of the Ra = 106 and
Pr = 7, associated with the long-term solution vorticity field in Fig. 11.
We have used E-max and k-max maps to synthesize data over several scales. This allows
us to study the effect of a range of scales with datasets that are easier to store and
visualize. Fig. 14 shows E-max and k-max maps for the vorticity field shown in Fig. 11.
The maximum energy is computed over scales of k = 15 to k = 20, which corresponds to a
resolution varying from 3.0 to 1.2% of the horizontal box width. The E-max map is very
similar to the small-scale scalogram slice in Fig. 13. However, only the highest values
near the shear layers are revealed. This simplifies the pattern and makes it easier for us to
pick out the most significant features, the shear layers. The lower energy features within
the convection cells are not shown on the E-max map. In Ra = 106 and Pr = 7 simulation
shown here, the pattern is relatively simple and many of the features highlighted in the
E-max map can be easily picked out on the small-scale scalogram slice. The ability of
the E-max to reveal the highest energy features and simplify the pattern will become
increasingly important as the convection patterns become more complicated with
increasing Rayleigh and Prandtl number.
Extremely large values of k-max (Fig. 14) occur along the edges of sharp gradients.
These areas are associated with differential movement of fluid. Identifying and
understanding areas of fluid movement is essential to growing metallic alloys from a
crystal mush since fluid movement within the mush can cause areas of anomalous
composition or freckles. The movement of fluid in a crystal mush is also important to
crystallization in geologic settings such as magma chambers, ice crystallization, and the
inner/outer core boundary (Beckermann, 2000). Material scientists have previously
relied on studies of the local Rayleigh number to identify regions of strong fluid
movement (Beckermann, 2000). The k-max has the added advantage of showing
movement both in the horizontal and vertical. The k-max not only picks out areas of
current movement, but also delineates where movement has occurred recently. It marks
the path of the fluid throughout the box.
Fig. 14. E-max and k-max maps for the
vorticity computed from k = 15 to k = 20 for Ra = 106 and Pr = 7 for the temperature
deviation field shown in Fig. 11.
Effect of Increasing Prandtl Number up to 100
For the Ra = 106 simulations we increased the Prandtl number from 7 to 17, 40, 80, and,
finally, 100. An example of a fluid with Pr = 17 is ethyl alcohol at 15oC and 1 atm
(Turcotte, 1982). Fluids at higher Prandtl numbers can be seen as a solution of water with
glycerin at 20oC and 1 atm (Weast, 1987). For example, Pr = 40 is a solution of water
and 48 vol% glycerin. Pr = 80 is a solution of water and 60 vol% glycerin. Pr = 100 is a
solution of water and 64 vol% glycerin. As the Prandtl number is increased, Eq. 2
becomes stiffer in time integration. The solution becomes more difficult to solve because
many more timesteps are needed before the initial temperature and velocity perturbations
cause convective movement. The overall time development of the solution evolves
slower. We also found that, as the Prandtl number was increased from Pr = 7, sharper
temperature gradients and stronger shear flows, i. e., velocity boundary layers developed.
This required using more grid points and smaller timesteps. This further increased the
computation time required for the simulations.
The most noticeable effect of even a small in increase in Prandtl number from 7 to 17 at
Rayleigh number 106 is the increase in horizontal shear flow (Fig. 15). This results in a
"wind" driving the plumes. This shearing effect due to inertia could be substantial in a
system such as Europa where there is an icy "lithosphere" being deformed by a
convecting, brine-rich, global ocean. If the dissolved species in the ocean and the
formation of ice increase the Prandtl number of the system, estimates on the behavior of
the system based on studies at Prandtl number 7 may not be appropriate, and we must use
higher Prandtl numbers going upwards to 50 or 100.
Fig. 15. Long-term solution of the
temperature deviation and vorticity fields for Ra = 106 and Pr = 17.
The smaller scale structures resulting from the strong horizontal shear flow makes it
difficult to pick out the number of convection cells from the vorticity field in Fig. 15.
This a situation where the wavelet transform proves especially valuable. In the large
scale vorticity scalogram slice (Fig. 16), we can easily see that the same pattern of two
wide convection cells separated by two narrow ones that we saw in simulations for Ra
=106 and Pr =7 (Fig. 13). The features related to the shear are discerned clearly in the
medium and small-scales.
Fig. 16. Scalogram of the Ra = 106 and
Pr = 17, associated with the long-term solution vorticity field in Fig. 15.
The E-max map for the smaller scales of the of the vorticity (Fig. 17) gives a similar
picture to the small-scale scalogram slice. The E-max, however, shows that the strongest
signatures are coming from two layers of circulation cells, one in the top of the fluid and
one in the bottom. The horizontal shear in this fluid appears to be creating a layered
convection style in the smaller scale. Vortex cells with positive sense of circulation have
much stronger signatures than those with negative sense of circulation. The original
vorticity field (Fig. 15) also shows stronger positive vortices than negative ones. The
k-max map (Fig. 16) emphasizes both the vertical shearing movement of the fluid along
with its significant horizontal component. The ability to pick out both the horizontal and
vertical movement within the time-dependent fluid motions provides an advantage over
the more traditional local Rayleigh number approach, which only emphasizes the vertical
movement (Zaleski, 2000).
Fig. 17. E-max and k-max maps for the
vorticity computed from k = 15 to k = 20 for Ra = 106 and Pr = 17 for the temperature
perturbation field shown in Fig. 15.
As the Prandtl number is increased to 100, the lateral temperature distribution shows
more layered structures. This indicates that there is better homogenization of the total
temperature field within each layer (Fig. 18). Both the hot and cold thermal boundary
layers decrease in thickness. The plume heads still spread laterally across the hot and
cold boundary layers, but they have less vertical thickness. This indicates the possibility
that the Prandtl number may have an important control over the vertical heat flux in
finite, but large Prandtl number convection.
Fig. 18. Long-term solution of the
temperature deviation and vorticity fields for Ra = 106 and Pr = 100.
The vorticity field undergoes noticeable changes with increasing Prandtl number (Fig.
18). Individual circulation cells are delineated weakly by strong vertical shear layers
along their edges. The large-scale slice of the vorticity scalogram shows that there are
only two convection cells (Fig. 19) rather than the four seen for smaller Prandtl numbers.
One of the convection cells is 1.5 times wider than the other. Although at first glance it
may appear that the scalogram shows three convection cells, this is due to the fact that the
continuous convection field has been cut along the box edges. We see clearly that there
are actually two convection cells in the medium scalogram slice, where one can see the
shear layers along the vertical edges of the cells. The medium and small scale scalogram
slices show sharp vertical shear layers along the larger convection cell boundaries along
with smaller convection cells embedded within the larger ones.
Fig. 19. Scalogram of the Ra = 106 and
Pr = 100, associated with the long-term solution of the vorticity field in Fig. 18.
The E-max map (Fig. 20) for the smaller scales portrays areas of sharp vertical shear
layers, which coincide with long narrow circulation cells seen in the medium-scale
scalogram slice. These long narrow cells are found in between the larger scale
circulation cells. The k-max (Fig. 20) emphasizes the primarily vertical nature of fluid
motions; although it shows that there is also some degree of horizontal movement.
Fig. 20. E-max and k-max maps for the
vorticity computed from k = 15 to k = 20 for Ra = 106 and Pr = 100 for the temperature
deviation field shown in Fig. 15.
Effect of Increasing Rayleigh Number
Simulations were run at a Rayleigh number of 108 for Prandtl numbers of 7, 17, 40, 60,
80, and 100. The increase in Rayleigh number leads to development of smaller scale
structures (Animation 2). The convection evolves through the branching of plumes and
the division and recoalescence of vorticity cells. The plumes become thinner than for Ra
= 106. The temperature deviation is more layered, thus indicating a greater separation of
the convective field. The shear layers along the edges of the convection zones are much
sharper.
Animation 2. Temporal Evolution of Ra = 108 and Pr = 7
simulation for an aspect-ratio of 3.
Click on picture below to start animation.
For Ra = 108 and Pr = 7, the convection pattern in the long time regime (Fig. 21) is
similar to the pattern for Ra = 106 and Pr = 100 (Fig. 18). There is one upwelling and
one downwelling, both of which travel across the complete vertical distance between the
horizontal boundary layers. The vorticity is characterized by two circulation cells with
one cell 1.5 times larger than the other. The vorticity field is also characterized by many
small-scale features deriving from the larger convection cells.
Fig. 21. Long-term solution of the
temperature deviation and vorticity fields for Ra = 108 and Pr = 7.
By comparing the temperature deviation scalogram for convection in a Prandtl number 7
fluid at Ra = 108 (Fig. 22) to that in a fluid with Ra = 106 (Fig. 12), we can see the
variations of the scales associated with the plumes. For Ra = 106, the general shape of
the plumes is visible in the medium-scale. In the medium-scale for Ra = 108, we can see
hot regions in the boundary layer where the plumes impinge, but there are only faint
traces of the plume stems in the medium-scale. The full details of the plumes are visible
only at the smaller scale for Ra = 108 .
Fig. 22. Scalogram of the temperature
deviation for the Ra = 108 and Pr = 7, associated with the long-term solution of the
temperature deviation field in Fig. 21.
Wavelet analysis becomes invaluable in analyzing the complicated vorticity fields
produced in higher Rayleigh number convection (Ra = 108) (Fig. 23). In the large-scale
scalogram slice we can see the structure of the two large convection cells. The
medium-scale slice gives us the smaller convection cells within the larger cells; while the
small-scale slice shows the small-scale shear layers lying along the edge of the
large-scale circulation cells.
Fig. 23. Scalogram of the Ra = 108 and
Pr = 7, associated with the long-term solution of the vorticity field in Fig. 21.
Since a large portion of the activity in this vorticity pattern occurs at the smallest scales,
it is useful to examine the E-max over the small-scales, from mode k = 15 to k = 20 (Fig.
24). The plume signatures that were also visible in the small-scale scalogram slice are
highlighted along with he vortices within the larger convection cells. These vortices
occur over a range of smaller scales. The k-max map (Fig. 24) shows a finer pattern than
found in the k-max maps for the simulations at Ra = 106. The vertical and horizontal
movements associated with the plume and the downwelling coexist with the movement
associated with the smaller vortices within the two larger circulation cells.
Fig. 24. E-max and k-max maps for the
vorticity computed from k = 15 to k = 20 for Ra = 108 and Pr = 100 for the temperature
deviation field shown in Fig. 21.
The temperature and vorticity fields at Ra = 108, Pr = 17 and 40 are somewhat similar to
those for Ra = 108, Pr = 7. When the Prandtl number is increased to 60 at Ra = 108,
however, the convection style changes dramatically (Fig. 25). The temperature
deviation field becomes more layered, indicating better homogenization of the
temperature within each layer. The vorticity field has become quite complicated, and it is
difficult to interpret these fields without the use of wavelets.
Fig. 25. Long-term solution of the
temperature deviation and vorticity fields for Ra = 108 and Pr = 60.
The change in the convection style when the Prandtl number is increased from 7 to 60
becomes clearly evident in the temperature deviation scalogram (Fig. 26). We do not see
any signatures from plumes or downwellings in the large-scale. In the medium-scale, we
only see the boundary layers. The boundary layer signatures are broken by numerous
small upwellings and downwellings. The shape of these upwellings and downwellings is
only apparent in the small-scale. We can see that these features do not traverse vertically
through the center of the box. A layered convection style has developed with an
additional internal boundary layer across the center of the box.
Fig. 26. Scalogram of the Ra = 108 and
Pr = 60, associated with the long-term solution of the temperature deviation field in Fig.
25.
The large-scale slice of the vorticity scalogram shows that large-scale circulation cells
still exist at this mediumly large Prandtl number (Fig. 27). There are four circulation
cells, which are much narrower than the large-scale circulation cells at lower Prandtl
numbers. The medium-scale scalogram slice shows a complex structure of circulation
cells within the large-scale circulation cells. The small-scale slice of the vorticity
scalogram is almost identical to the small-scale slice of the temperature deviation
scalogram (Fig. 26) . There are areas of downwellings and upwellings associated with
sharp shear layers that travel vertically to the center of the box. Large plumes that travel
through the vertical distance of the box no longer exist. We note that this change in
convection style is brought about by increasing the Prandtl number, while keeping the
Rayleigh number constant. This indicates that the Prandtl number plays a significant role
in determining the convection style. The relative increase in viscosity to thermal
diffusivity as the Prandtl number increases makes the movement of large-scale plumes
and downwelling more difficult. More of the heat appears to be transported by smaller
scale plumes at higher Prandtl numbers.
Fig. 27. Scalogram of the Ra = 108 and
Pr = 60, associated with the long-term solution vorticity field in Fig. 25.
The E-max of the vorticity field shows the location of the shear layers of the stronger
plumes and downwellings (Fig. 28). The heads of the plumes and downwellings have
especially strong signatures. We also see the layered structure of the convection with no
plumes crossing the entire box. The k-max map for Ra = 108 and Pr = 60 is quite
different from that of Ra = 108 and Pr = 7. We can see evidence of a large amount of
horizontal movement, but only limited vertical movement. Although many of the
structures do not cross the center of the box, it appears that there is still some movement
across the center. This indicates that the convection style is not completely layered.
Fig. 28. E-max and k-max maps for the
vorticity computed from k = 15 to k = 20 for Ra = 108 and Pr = 60 for the vorticity field
shown in Fig. 25.
The convection style becomes more distinctively layered for Ra = 108 and Pr = 80. The
scale of the structures also becomes smaller, and the convection appears to have a more
chaotic character. The features just discussed become quite well defined for Ra = 108
and Pr = 100 (Fig. 29). The temperature deviation and vorticity fields are quite
complicated, and the scalogram technique is required for their interpretation.
Fig. 29. Long-term solution
temperature deviation and vorticity fields for Ra = 108 and Pr = 100.
The slice showing the vorticity scalogram for the large-scale slice (Fig. 30) displays that
there are still some large circulation cells. There are basically two layers of large-scale
cells; the convection style has become indisputably layered. The convection cells are
tilted, indicating the presence of horizontal shear. The medium-scale scalogram slice
shows a complicated pattern of narrow, tilted convection cells. This confirms the
presence of a horizontal shear. The small-scale slice shows the small shear layers created
by individual plumes. We can see that the plumes fade away when they reach about
halfway through the box height. A new boundary layer has been formed across the
middle of the box by the presence of layered convection.
Fig. 30. Scalogram of the Ra = 108 and
Pr = 100, associated with the long-term solution of the vorticity field in Fig. 29.
The small scalogram slice is complicated for this simulation. Although it only looks at k
= 20, the amount of small-scale features in this vorticity field indicates that other small
scales should play an important role. We have examined the E-max over scales ranging
from k= 15 to 20 (Fig. 31). In this image we can see both the long narrow shear layers
along the edges of larger convection cells and the smaller scale vortices that are further
subdividing the vortices observed in the medium-scale. The vortices with scales ranging
from medium to small indicate that although the convection still has some large-scale of
organization, the smaller scales are becoming increasingly important. The k-max (Fig.
31) emphasizes the chaotic movement throughout the box. There is less movement
across the center of the box than for Ra = 108 and Pr = 60. This indicates that the
convection style is reaching a truly layered state for this intermediately high Prandtl
number situation.
Fig. 31. E-max and k-max of Ra = 108
and Pr = 100 for k = 15 to 20 of the long-term solution of the vorticity field in Fig. 29.
Reynolds Number
The turbulent Reynolds number was calculated a posteriori for several of the simulations
(Table 3). The Reynolds number was calculated by post-processing the data from the
run. The Reynolds number was not calculated from an assumed Rayleigh and Prandtl
number relationship.
The Reynolds number is defined as:
(8)
where V is the dimensional root-mean-square velocity, l is the dimensional velocity, and
is the dimensional kinematic viscosity.
We calculated the Reynolds number a posteriori as:
(9)
V 0 is the nondimensional root-mean-square or quadratic mean velocity. It represents the
velocity of the largest eddies. l0 is the nondimensional integral length scale or
autocorrelation length. It represents the size of the largest eddies. V 0 and l0 are local
quantities that are calculated in the course of the run. Since V 0 and l0 are
nondimensionalized by the thermal diffusivity and the box height, the resulting Reynolds
number is that defined in Eq. (8).
Rayleigh Prandtl
Number number
Aspect-Ratio
106
40
3
Reynolds
Number*
2.2
106
80
3
0.63
106
100
3
0.45
Table 3. Reynolds numbers for several simulations. * indicates that the Reynolds
numbers were calculated a posteriori after each run.
The Reynolds numbers tended to decrease with Prandtl number for Ra = 106, which is in
the low Reynolds number regime for these Prandtl numbers. There are two effects
contributing to the decrease in Reynolds number with Prandtl number. The first is the
direct effect of dividing the quantities V 0 and l0 by the Pr. The second effect is the that
V 0 * l0 tended to decrease with Prandtl number. The smaller size of the eddies with
increasing Prandtl number was also picked up visually on the scalogram (e.g. Fig. 19 &
30).
Three-Dimensional Convective Effects
The systems discussed in this study are all two-dimensional. In 2-D systems the vorticity
vector is always perpendicular to the flow plane. This means that no vortex stretching
can occur in 2-D flows since the vorticity cannot be amplified by the velocity gradient.
Vorticity can only be weakened by viscous dissipation. A fundamental difference in the
nature of turbulence in 2-D and 3-D flows is that energy in large vortices cannot be
transported to smaller scale vortices through vortex stretching; instead in 2-D energy at a
given scale is transferred to larger scales and dissipated at the system boundaries
(Belmonte, et al., 2000). Due to the absence of vortex stretching, it is more likely that
fluctuations will be transported downstream intact in 2-D than in 3-D flow (Belmonte, et
al., 2000). Lateral mixing effects that can remove and introduce fluctuations would also
be weaker in 2-D than in 3-D flow (Belmonte, et al., 2000).
Three-dimensional effects have important influences on geophysical systems. The
vertical velocity introduced by buoyancy in atmospheric flows induces vortex stretching
that is responsible for the spiraling updrafts and downdrafts seen in thunderstorms and
dust whirls (Cortese and Balachander, 1993). For systems such as Europa’s global ocean,
rotation caused by Coriolis effects may have limit how far plumes can spread laterally
(Thomson and Delaney, 2001). One type of flow that is characterized by the formation
of cigar-shaped vortices through vortex stretching is known as ABC flow (Aref and
Brøns, 1998, Dombre et al., 1986). The stretching ability of these flows leads to their
ability to amplify magnetic fields and form dynamos (Galloway and Frisch, 1986). This
type of flow has been suggested as a mechanism for dynamo production in the Sun
(Dorch, 2000). Vortex stretching flows also are responsible for dynamos produced in the
outer core of the Earth and the oceans of the Jovian moon Callisto and Europa (Khurana,
et al., 1998). We will conduct future studies on 3-D finite Prandtl flows to further
enhance our understanding of these important flows in geophysical systems.
Schmalzl et al. (2001) found that 3-D effects are most important in flows with Prandtl
numbers below 3 when Ra = 106 due to the fact that the toroidal flow, which is neglected
in 2-D flow. They found two flow regimes. In the regime at Prandtl numbers below 3,
rising currents are distorted by toroidal currents leading to distortion of currents and
exchange of heat between plumes and the interior. This results in diffusion being the
main mechanism of heat flux. As the Prandtl number increased, the toroidal flow
decreased, and most of the heat was transported by plumes through the isothermal interior
to thermal boundary layers.
The simulations of Schmalzl et al. (2001) were done at Ra = 106. Rescaling the
convection equations by the free fall velocity rather than the thermal diffusivity gives a
better understanding of the effect of increasing the Rayleigh number on the toroidal
flow. The free fall velocity is given by:
(10)
where is the coefficient of thermal expansion. The conservation of momentum
equation nondimensionalized by the free fall velocity is:
(11)
As the Rayleigh number increases, the diffusional term will drop out of the conservation
of momentum equation for Pr/Ra < 10-8. The change in poloidal velocity with time can
then be expressed as:
(13)
and the change in toroidal velocity in time can be expressed as:
(14)
The toroidal velocity, therefore, should not have a significant effect at Rayleigh numbers
of about 108 for Prandtl number of around 3. In short, the toroidal motions should be
saturated at high Rayleigh numbers under the conditions prescribed in Eqs. 13 and 14.
Conclusions
We have found that at a given Rayleigh number, increasing the Prandtl number can cause
the convection style to change from non-layered to layered. Other effects of increasing
the Prandtl number at a given Rayleigh number include an increase in the number of
plumes, a thinning of the plumes and boundary layers, and an increase in the horizontal
shear. The overall size of the large eddies tend to decrease with Prandtl number. This is
partially responsible for the decrease in Reynolds number with Prandtl number.
As the Rayleigh number and Prandtl number were increased, the vorticity fields became
more complicated. The wavelet transform helped us interpret these fields by allowing us
to separate out the processes occurring at different scales without losing the specific
spatial location of the signals (van den Berg, 1999). This allows us to see the overall
large-scale convection style even when it is masked by small-scale structures. The large
scale scalogram slices were used to pick out large-scale circulation cells and layered
structures. The small-scale scalogram slices allowed us to separate out smaller scale
processes, such as shear zones along circulation cell edges and smaller vortices within
larger circulation cells.
The E-max and k-max distributions were used to synthesize the data in the scalogram
(Bergeron, 2000). The E-max gives the energy for the scale that has the maximum
energy over a given range of scales. It allows one to quickly find the most prominent
feature in that scale range. We found it useful for quickly picking out features, such as
shear zones at the smaller scales where there was often signals from many sources. It
was especially helpful for simplifying vorticity datasets in simulations at higher Rayleigh
and Prandtl numbers where a large number of vortices occurred over a wide range of
scales.
The k-max maps out areas of sharp gradients. It was used to draw a path of fluid
movement through the box. The k-max could be used in simulations of crystal mushes to
pick out areas of movement where anomalous compositions might occur in the solidified
fluid. Identifying areas of possible anomalous compositions is important in the casting of
metallic alloys and may also prove useful in modeling flow in crystal mushes in magma
chambers, the inner and outer core boundary, and sea ice (Beckermann, 2000).
We have found four major styles of convection in the long-term solutions for simulations
with an aspect-ratio of 3. For Ra =106 and Pr = 7 to 80, there are four convection cells:
two wide cells separated by two narrow cells. For Ra =106 and Pr = 100 and Ra =108
and Pr = 7 to 17, there are two convection cells with one cell that is about 1.5 times
wider than the other cell. At Prandtl number 60 there are 4 narrow convection cells. A
major change in convection style is evident at Pr = 60. The increasing relative viscosity
results in more heat being transported by small-scale thermal plumes than by large-scale
plumes. At Ra =108 and Pr = 100 the four narrow cells divide into two layers of cells,
and the convection style becomes layered. This study indicates that the convection style
is dependent on both the Rayleigh number, Prandtl number. We will conduct future
studies in 3-D to help determine the effect of the third dimension on these flows.
Since increasing the Prandtl number can greatly increase the shear in the fluid, the
Prandtl number of the fluid may have significant effects on the type of deformation seen
in a "lithospheric" layer above a convecting finite Prandtl fluid. For example a large
shear force created in the convecting brine-rich ocean of Europa (Carr et al., 1998,
Khurana et al., 1998) could explain some of the surface deformation seen on this Jovian
moon.
We have also found that an increase in Prandtl number can have a large influence on both
the thickness of the boundary layers and the overall convection style, the scale at which
plumes develop, the number of convection cells, and whether or not convection is
layered. Wavelets have allowed us to analyze the convection simulations at different
scales to better understand these changes in convection style. The change in convection
style with Prandtl number indicates that the Prandtl number has a significant effect on the
heat flux in the fluid. We have seen that the Prandtl number should be important in the
Nusselt number scaling (Grossmann and Lohse, 2000, Zaleski, 2000, Verzzico and
Casmussi, 1999, Shairman and Siggia, 1990, Castaing et al., 1989). Future studies will
examine quantitatively the dependence of the Nusselt number on the Prandtl number. A
better understanding of heat flux in finite Prandtl number convection will lead to a better
understanding of the heat budget of planetary bodies with convecting global oceans such
as the Jovian moon Europa. It will also allow a better understanding of heat flux on the
early Earth during the magma ocean period. The style of convection in this magma ocean
may also be important to the chemical differentiation of the early Earth.
Acknowledgments
We would like to acknowledge Alain P. Vincent, David Munger, Ludek Vecsey, and
Fabien Dubuffet for their help with this study. Catherine Hier Majumder thanks the NSF
Fellowship program for providing support during this study. This research was also
supported by the Complex Fluids Program of D. O. E.
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