Nonlinear viscous fluid patterns in a thin rotating spherical domain

Nonlinear viscous fluid patterns in a thin rotating spherical domain
and applications
Ranis N. Ibragimov
Citation: Phys. Fluids 23, 123102 (2011); doi: 10.1063/1.3665132
View online: http://dx.doi.org/10.1063/1.3665132
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Published by the American Institute of Physics.
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PHYSICS OF FLUIDS 23, 123102 (2011)
Nonlinear viscous fluid patterns in a thin rotating spherical domain
and applications
Ranis N. Ibragimov
Department of Mathematics, University of Texas at Brownsville, Brownsville, Texas 78520, USA
(Received 6 September 2011; accepted 7 November 2011; published online 9 December 2011)
We study the nonlinear incompressible fluid flows within a thin rotating spherical shell. The model
uses the two-dimensional Navier-Stokes equations on a rotating three-dimensional spherical
surface and serves as a simple mathematical descriptor of a general atmospheric circulation caused
by the difference in temperature between the equator and the poles. Coriolis effects are generated by
pseudoforces, which support the stable west-to-east flows providing the achievable meteorological
flows rotating around the poles. This work addresses exact stationary and non-stationary solutions
associated with the nonlinear Navier-Stokes. The exact solutions in terms of elementary functions for
the associated Euler equations (zero viscosity) found in our earlier work are extended to the exact
solutions of the Navier-Stokes equations (non-zero viscosity). The obtained solutions are expressed
C 2011 American Institute of Physics.
in terms of elementary functions, analyzed, and visualized. V
[doi:10.1063/1.3665132]
I. INTRODUCTION
The large-scale atmospheric dynamics is usually described
by latitude-dependent viscous or nonviscous three-dimensional
flows within the theory of shallow water approximation,1–5,7,11
where the surface is considered to be free and the depth of the
atmosphere layer obeys an additional equation of motion. The
modelling of such moving air masses plays an important role in
understanding the impacts of climate change.3,8,9
We address the nonlinear three-dimensional Euler (zero
viscosity) and Navier-Stokes (nonzero viscosity) equations
for an incompressible fluid in the Lamb’s form,10
~
@~
u
u~
u p
~ þ D~
~
u rotu ~
u ¼ r
þ þ W þ 2~
uX
u;
@t
q0
2
(1)
r~
u¼0
in a thin spherical shell
P ¼ x 2 R3 : r0 < jxj < r0 þ ;
(2)
(3)
where ! 0 is a small parameter for the thickness of the
spherical shell, ~
u is the velocity vector, p is the pressure pressure, q0 is the constant density, is the kinematic viscosity,
and W is the potential of the gravitational force which
~ is
includes the Newtonian attraction. The term 2~
uX
referred to as the Coriolis acceleration. Additionally, r0 is a
fixed altitude above the earth.
Since the velocity vector ~
u and the pressure p are coupled
together by the incompressibility constraint r ~
u ¼ 0; it is
difficult to analyze the full set of three-dimensional Eqs. (1)
and (2). A common approach to simplify the nonlinear model
in the question is to use the artificial methods such as the pressure stabilization and projection.11 As has been shown in Ref.
11, the error estimate of the pressure stabilization and projection methods is not however mathematically precise. Instead,
we employ the usual spherical coordinates (r,h,/) and use the
1070-6631/2011/23(12)/123102/8/$30.00
reduction of the Eqs. (1) and (2) to the two-dimensional equations on the sphere
S ¼ fðh; /Þ;
0 < h < p;
0 / < 2pg;
(4)
which is obtained in the limit ! 0. In our modeling, the
depth is sent to zero in the limiting procedure for the non-free
surface model. The latter reduction relies on the theorem from
the work Temam and Ziane.12 Namely, as follows directly
from our previous analysis in Refs. 13 and 16, the latter reduction is justified by theorem B in Ref. 12, which states that the
strong global solution of the three-dimensional Navier-Stokes
equations (1) and (2) converges as ! 0 to the strong global
solution ~
uðh; /; tÞ of the two-dimensional Navier-Stokes or
Euler equations on a spherical surface (4), where
~
vðh; /; tÞ ¼ lim
1
!0 r0
r0 þr0
r0
r~
uðr; h; /; tÞdr ¼ 0; vh ; v/ ;
(5)
in which r0 is constant and is a nondimensional small parameter. The vector ~
vðh; /; tÞ can thus be interpreted as the
average velocity with respect to r.
In terms of physical interpretation, the above reduction
can be interpreted as follows: let the earth be a sphere of
radius r0. Then, for r ¼ r0, we must set ur ¼ 0. In addition,
for r ! 1, we must assume that (qur) ! 0 (the atmosphere
neither loses nor gains mass from outside). However, there
exists a height for which qur attains a maximum (the other
possibility corresponds to the case when the vertical velocity
ur ¼ 0), i.e., where
@qur ¼ 0:
@r r¼r0
It is thus postulated in the present work that at least one such
level exists for the entire atmosphere of the earth. We call it
the "mean level." In practice (see e.g., Refs. 2 and 14), we
can speak about it as a level of 3–5 km. Thus, in the limit
23, 123102-1
C 2011 American Institute of Physics
V
123102-2
Ranis N. Ibragimov
Phys. Fluids 23, 123102 (2011)
! 0, the non-stationary three-dimensional viscous rotating
fluid flow is confined on a sphere (4) of the radius r0 parametrized by the polar (latitude) angle h and azimuthal (longitude) angle /.
Recently, the flow of a thin layer of incompressible fluid
on a rotating sphere has been studied in Ref. 9 in order to
investigate a linearized theory for small amplitude perturbations about a base westerly flow field, allowing calculation of
the linearized progressive wavespeed. The authors of Ref. 9
have extended their result to the numerical solution of the full
model in the absence of viscosity, to obtain highly non-linear
large-amplitude progressive-wave solutions in the form of
Fourier series. It is shown in Ref. 9 that the formation of localized low pressure systems cut off from the main flow field is
an inherent feature of the non-linear dynamics, once the amplitude forcing reaches a certain critical level. Our previous
analysis in Ref. 15 extends the earlier results in Ibragimov and
Pelinovsky13,16 to the case of nontrivial and non-steady nonlinear solutions in the particular case of zero viscosity. This
work is an extension of the generalized results in Ref. 15 to
the case of nonzero viscosity. Particularly, here, we focus on
integration of the nonlinear non-steady latitude-dependent
Navier-Stokes equations in a thin rotating spherical shell. The
inquiry is motivated by dynamically significant Coriolis forces
in meteorology and oceanographic applications such as
climate variability models, the general atmospheric circulation
model, weather prediction (see e.g., Refs. 17–21), as well as a
variety of applications to large-scale dispersant operations
e.g., Deepwater Horizon incident studied recently in Ref. 22.
II. PRELIMINARIES
As follows directly from our previous work in Refs. 13
and 16, in the particular case of no rotation, there exists an
exact stationary solution to the three dimensional NavierStokes equations in spherical coordinates given by
ur ¼ 0;
uh ¼
a
;
r sin h
u/ ¼ 0;
p¼b
a2
; (6)
2r 2 sin2 h
where (a, b) are arbitrary parameters. The stationary solution
(6) describes the flow tangential to a sphere of any given
radius r. The stationary flow has two pole singularities at
h ¼ 0 and h ¼ p. These singularities correspond to the source
and sink of the velocity vector at the North and South poles
of the spherical shell P: the fluid is injected at the North
pole from an external source and it leaks out at the South
pole to an external sink. It has been shown in Ref. 16 that the
steady-state solution (6) is asymptotically stable with respect
to symmetry-preserving perturbations for all possible Reynolds numbers.
If the effects of rotation are ignored, the exact solution
(6) corresponds to the exact stationary solution of the twodimensional Navier-Stokes equations with nonzero viscosity
on the unit sphere S
vh ¼
a
;
sin h
v/ ¼ 0;
where (a, b) are arbitrary parameters.
p ¼ b;
(7)
However, the exact solution (7) is destroyed by the
effects of rotation, whereas another stationary solution
ð
F
1
vh ¼ 0; v/ ¼ FðhÞ; p ¼ p ¼ F cos h
dh þ p0 ;
þ
sin h R0
p0 ¼ const:
(8)
persists with respect to the rotational terms provided that
F¼
1
sin h
(9)
if the effects of viscosity are included. We remark that the
both stationary flow solutions have singularities in the pressure, which cannot be removed by a coordinate transformation. Particularly, in the spherical geometry, the curvature of
the shape induces singularities in the velocity vector and in
the pressure terms. If the spherical layer is transformed to a
planar layer by a homotopy transformation, the stationary
flows with singularities transform to a regular solution with a
constant velocity field.
III. NONLINEAR NON-VISCOUS FLOWS IN
NON-ROTATING REFERENCE FRAME
The presence of Coriolis effects originates the general
west-to-east flows caused by the Earth’s rotation which are
related to jet streams, i.e., zones of fastmoving west-to east
winds in the upper atmosphere between the Ferrell and Polar
cells. Such flows exist due to the presence of the cold North
and South poles and warm equator so that the pressure is low
at high latitudes above the poles and is high in the temperature zones and even higher in the equatorial zones. Thus, as
has been discussed in Ref. 23, under the assumption of no
friction, and a distribution of temperature dependent only
upon latitude and altitude, the stationary solution (8) can be
associated with a zonal flow directed from the west to east.
In view of the importance of zonal flows in terms of
meteorological applications,14,23,24 we first focus on the system (1) and (2) on the sphere S in the limiting case Re ! 1
(nonviscous flows) and Ro ! 1 (no rotation), and with the
superimposed stationary flows of the form (8) with F being
an arbitrary function of h.
In particular, as it has been remarked in Ref. 23, under the
assumption of no friction and a distribution of temperature
dependent only upon latitude, starting from some height, the
west-to-east flows always exist in the earth’s atmosphere if the
atmosphere overtakes the earth in the west-to-east motion. The
possible modeling scenario when a steady condition of motion
can occur in the atmosphere with an unvarying distribution of
temperature (i.e., atmospheric movement that remains constant
at each individual point in the atmosphere) has also been discussed in Ref. 23. The experimental results in Ref. 24 justify
many of the conclusions discussed in Ref. 23.
We thus look for solution in the form
vh ;
vh ¼ l^
v/ ¼ FðhÞ þ l^
v/ ;
p ¼ p þ l^
p;
(10)
where l > 0 is a parameter, v^h ; v^/ , and p^ are spatial and time
dependent disturbances of the basic flow (8). Since we are
123102-3
Nonlinear viscous fluid patterns
Phys. Fluids 23, 123102 (2011)
interested in exact solution of the nonlinear model, the case
l 1 is of special interest. The particular case l 1 was
studied in much details in our previous work in Refs. 13 and
16. Thus, after the above averaging procedure, in the double
limiting case when Re ! 1 and Ro ! 1, the basic model is
given by means of the perturbed Euler equations of motion
of inviscid fluid in non-rotating reference frame as (hereafterm, the symbol hat is omitted and, without loss of generality, the parameter l is set to be one)
@vh
F @vh
@p
@vh
þ
2Fv/ cot h þ
þ vh
@t sin h @/
@h
@h
v/ @vh
þ
v2/ cot h ¼ 0;
sin h @/
(11)
(13)
where prime means differentiation.
The pressure terms can be eliminated in order to write
the Euler equations (11)–(13) as a single equation in terms of
the stream function w(t,h,/) as
@DS w
F @DS w w/ L1 F
1
þ
J ðw; DS wÞ ¼ 0; (14)
þ
@t
sin h @/
sin h
sin h
where
1 @w
@w
; v/ ¼
;
sin h @/
@h
1 @
@
1 @2
DS ¼
sin h
þ 2
sin h @h
@h
sin h @/2
vh ¼ (15)
is the Laplace-Beltrami operator in spherical angles and
1 d
d
1
(16)
sin h
2
L1 ¼
sin h dh
dh
sin h
is the Stourm-Liouville operator for the associated Legendre
functions. Additionally,
J ða; bÞ ¼ ah b/ a/ bh
in Eq. (14) stands for the nonlinear Jacobian operator with
the subscripts meaning the partial differentiation.
The model (14) corresponds to the Euler model perturbed by zonal flows (8) in the non-rotating reference frame
with an arbitrary choice of F(h). It has been shown in our
recent work in Ref. 15 that solution of determining equations
(see e.g., the Eq. (6.16) in Ibragimov and Ibragimov25) provides the following l–independent two invariant solutions
c1 c2 h
1
½1
(17)
w ¼ þ lntan ; w½2 ¼ Uðk0 Þ;
t
t
2
t
where
and U is given in terms of elementary functions of its argument as
1n
;
(18)
UðnÞ ¼ c1 þ c2 ln
1þn
in which jk0j = 1 and c1 and c2 are arbitrary constants.
IV. NONLINEAR VISCOUS FLOWS IN ROTATING
REFERENCE FRAME
@v/
F @v/
1 @p
@v/
þ ðF0 þ cot hÞvh þ
þ
þ vh
@t
@h
sin h @/ sin h @/
v/ @v/
þ
þ vh v/ cot h ¼ 0;
(12)
sin h @/
@
@v/
¼ 0;
ðvh sin hÞ þ
@/
@h
k0 ¼ sin h cos /
The inclusion of the Coriolis force creates a cyclonic rotation around the poles, i.e., west-to-east winds.26 Namely, as
has been indicated in Ref. 23, the temperature difference
between the equator and the poles of a sphere gives rise to
waves of two kinds. The first kind consists of waves that
advance in the direction of the meridian; the second kind
includes waves that advance in the direction of the circles of
latitude. The atmospheric pressures and motions resulting
from the combination of these two groups of intersecting
waves give rise to the cyclonic and anticyclonic phenomena
which are nowadays a paramount topic of research in atmospheric modeling. For example, the mechanisms of cyclone formation in the Earth’s atmosphere have been recently studied in
Ref. 14 with the help of numerical modeling using the complete system of gas-dynamic equations. The authors of Ref. 14
have shown that cyclones can appear in horizontal stratified
shear flows of warm and wet air masses with a horizontal
direction of gradients of the wind velocity components as a
result of small disturbances of pressure which can be produced
by Rossby waves. Recent laboratory experiments on cyclone
and anticyclone formation in a rotating stratified fluid have
been studied in Ref. 24. The importance of understanding the
formation of cyclones and their time evolution in the Earth’s
atmosphere for the creation and distribution of weather systems throughout the world is summarized in Ref. 8.
In terms of applications to atmospheric sciences, it is
useful to note atmospheric patterns at the North and South
Poles spin around itself in the anticlockwise sense at a rate
X ¼ 2p rad/day, whereas atmospheric patterns in the domain
h [ [h0,p h0] do not spin around themselves but simply
translate provided h0 2 0; p2 . Owing to the Coriolis effects,
the achievable meteorological flows rotating around the
poles correspond to the flows that are being translated along
the equatorial plane. This translational motion is well captured by the zonal flows considered in our work.
We consider now the model Eq. (14) superimposed by
nonzero viscosity (Re = 1) and nonzero rotation (Ro = 1).
The corresponding two-dimensional Euler equations
(11)–(13) are then generalized to the Navier-Stokes equations in rotating reference frame as follows:
@vh
F @vh
v/ cos h
þ
2Fv/ cot h þ
@t sin h @/
Ro
@p
@vh
v/ @vh
þ
v2/ cot h
þ vh
þ
@h sin h @/
@h
1
vh
2 cos h @v/
þ
¼ 0;
DS vh 2 Re
sin h sin2 h @/
(19)
123102-4
Ranis N. Ibragimov
Phys. Fluids 23, 123102 (2011)
@v/ dF
F @v/
vh cos h
þ
þ Fvh cot h þ
vh þ
@t
dh
sin h @/
R0
1 @p
@v/
v/ @v/
þ vh
þ
þ vh v/ cot h
þ
sin h @/
@h sin h @/
1
v/
2 cos h @vh
;
þ
DS v / 2 þ
Re
sin h sin2 h @/
@
@v/
¼ 0:
ðvh sin hÞ þ
@/
@h
(20)
(21)
V. DISCUSSION AND VISUALIZATION OF INVARIANT
SOLUTIONS
We next introduce
t
k ¼ sin h cos / þ
2Ro
eral case of viscous flows on a rotating spherical surface, we
apply the method of approximate equivalence transformations (the general algorithm is assembled in Ibragimov and
Ibragimov25 and its application to a certain class of nonlinear
wave equation has been demonstrated in Ibragimov27). Our
main results on the extension of invariant solutions (17) are
summarized in Table I. The remainder of this article is
devoted to discussion and visualization of the tabulated
results.
½i
(22)
and denote H(h) for an integral of F, i.e., H0 (h) ¼ F(h). In
order to extend the invariant solutions (17) to the most gen-
½i
The non-trivial non-steady exact solutions vh and v/ for
the Euler equations (Re ¼ 1) were obtained in our previous
analysis in Ref. 15. Although the particular result for the
Navier-Stokes model (Re = 1) in a non-rotating reference
frame (Ro ¼ 1) was not reported in Ref. 15, one can check
TABLE I. Summary of invariant solutions and limiting cases. Since we are interested in the exact solutions of strongly nonlinear equations, we set l ¼ 1.
Rotation
Viscosity
Zonal flows
Invariant solutions
½1
vh ¼ 0;
[1]
Ro ¼ 1
Re ¼ 1
F¼0
[2]
Ro ¼ 1
Re = 1
F¼0
[3]
Ro = 1
Re ¼ 1
F¼0
½2
½1
0
c2
v/ ¼ t sin
h
vh ¼ U ðtk0 Þ sin /
0
v/ ½2 ¼ U ðtk0 Þ cos h cos /
c2
½1
½1
vh ¼ 0; v/ ¼
t sin h
U0 ðk0 Þ
½2
sin /
vh ¼
t
0
U ðk0 Þ
½2
cos h cos /
v/ ¼
t½1
c2
sin h
½1
vh ¼ 0; v/ ¼
t sin h 2Ro
U0 ðkÞ
t
½2
sin / þ
vh ¼
t
2Ro
U0 ðkÞ
t
sin h
½2
cos h cos / þ
v/ ¼
t
2Ro
2Ro
½1
c2
sin h
½1
v/ ¼
FðhÞ
t sin h 2Ro
0
U ðkÞ
t
sin / þ
¼
t
2Ro
U0 ðkÞ
t
sin h
cos h cos / þ
¼
FðhÞ
t
2Ro
2Ro
c2
½1
¼ 0; v/ ¼
FðhÞ
t sin h
U0 ðk0 Þ
sin /
¼
t
U0 ðk0 Þ
cos h cos / FðhÞ
¼
t
vh ¼ 0;
[4]
Ro = 1
Re ¼ 1
F ¼ F(h)
½2
vh
½2
v/
½1
vh
[5]
Ro ¼ 1
Re ¼ 1
F ¼ F(h)
½2
vh
½2
v/
½1
vh ¼ 0;
[6]
Ro ¼ 1
Re = 1
F ¼ F(h)
½2
½2
U0 ðk0 Þ
sin /
t
0
U ðk0 Þ
cos h cos / FðhÞ
¼
t
½1
c2
½1
v/ ¼
t sin h U0 ðkÞ
t
sin / þ
¼
t
2Ro
U0 ðkÞ
t
cos h cos / þ
¼
t
2Ro
vh ¼ 0;
Ro = 1
Re = 1
F¼
sin h
2Ro
c2
FðhÞ
t sin h
vh ¼
v/
[7]
½1
v/ ¼
½2
vh
½2
v/
123102-5
Nonlinear viscous fluid patterns
Phys. Fluids 23, 123102 (2011)
by direct differentiation that the invariant solutions of the
corresponding Navier-Stokes model are the same as for the
Euler model (Re ¼ 1) in non-rotating reference frame for arbitrary choice of F(h) (see also entry [6] in Table I). The crucial difference between the results obtained in Ref. 15 and
the present studies is that application of approximate equivalence transformations25,27 extends the invariant solution (17)
to the Navier-Stokes model (Re = 1) in a rotating reference
frame (Re = 1) as shown in the entry [7] of the Table I.
Namely, in terms of the stream function, the extended invariant
solution for arbitrary l > 0 can be written as
cos h H ðhÞ c1 c2 h
½1
þ þ lntan ;
w ¼
(23)
2lRo
l
2
t
t
w½2 ¼
cos h H ðhÞ 1
þ UðkÞ;
2lRo
l
t
(24)
where k is given by Eq. (22).
According to the entry [7], we remark that the extended
invariant solutions (23) and (24) are valid provided that F(h)
is constrained by the choice
F¼
sin h
:
2Ro
(25)
One can check by direct differentiation that the both w[1] and
w[2] given by Eqs. (23) and (24) solve the “viscous
corrections” of the Euler equations (see also Eqs. (19) and
(20), i.e.,
DS vh vh
2 cos h @v/
¼ 0;
sin2 h sin2 h @/
(26)
DS v/ v/
2 cos h @vh
þ
¼ 0;
sin2 h sin2 h @/
(27)
if F(h) is the solution of the non-homogeneous equation
L1 F ¼
sin h
:
Ro
It is also a matter of direct differentiation to check that F
given by Eq. (25) is a solution of the linear ordinary differential equation (ODE) (28).
We next visualize the obtained invariant solutions in
terms of the velocity and pressure fields associated with the
stream functions w[1] and w[2] given by Eqs. (23) and (24)
(see also the entry [7] in Table I). For example, the exact
solutions corresponding to the invariant solution w[1] for the
velocity field and the pressure are written in terms of elementary functions as follows:
1 @w½2
2c2
t
½ 2
;
¼ sin
/
þ
vh ¼ sin h @/
2Ro
t 1 k2
@w½2
2c2
t
½ 2
¼ cos
h
cos
/
þ
v/ ¼
@h
2Ro
t 1 k2
p½2 ¼ c2
4c22
;
R
ð
t;
h;
/
Þ
t2
2t2 ð1 k2 Þ
where
2
3
sin h
tan
6
t
2 7
7
Rðt; h; /Þ ¼ arctan6
4cot / þ 2R0 þ t 5
sin / þ
2R0
2
3
sin h
tan
t
6
7
4 2 5:
arctan4cot / þ
t
2R0
sin / þ 2R0
The exact solutions for the velocity field and the pressure
corresponding to the second invariant solution w[2] can be
written in terms of elementary functions likewise.
For the purpose of visualization, the solutions are plotted on a spherical surface using the symmetric interval
[z0, z0], where z0 ¼ cos h0, which corresponds to the truncated annular domain
S0 ¼ fðh; /Þ : h0 h p h0 ;
(28)
0 / 2pg;
(29)
where 0 < h0 < p/2. Without loss of generality, the spherical
layer S0 is truncated symmetrically at the two rings located
FIG. 1. (Color online) Comparison of exact
solutions from entry [7] in Table I written in
terms of the azimuthal velocity v/ component
plotted on a spherical surface at different values
of time t. The upper panel shows the invariant
½1
solution v/ at the initial time t ¼ 1 and latter
times of t ¼ 2, 6, 12, 24, and 40. The lowermost
panel is used to compare the invariant solution
½2
v/ at the same values if time. The constant
Ro ¼ 68.493.
123102-6
Ranis N. Ibragimov
Phys. Fluids 23, 123102 (2011)
FIG. 2. (Color online) Comparison of exact solutions
from entry [7] in Table I written in terms of the azimuthal velocity v/ component plotted on a spherical
surface at different values of time t. The circles along
the centerline (red online) correspond to the stationary
½IP
trivial solution v/ ¼ 1= sin h whose asymptotic stability was investigated in our previous studies in Ref. 16.
The dotted and dashed lines correspond to the invariant
½1
½2
solutions v/ and v/ , respectively, at / ¼ 50. The constant R0 is the same as in Figure 1 and values of t [
[0.5, 10].
in the northern and southern semispheres so that the invariant
solutions are free of pole singularities in S0.
The values of the constants were derived from the boundary conditions given by Eq. (42) in Ref. 16. Namely, the solution for the truncated spherical layer S0 given by Eq. (29)
is defined on the closed interval h0 < h p h0 for
h < h0 < p/2 is subject to the boundary conditions
w(h0) ¼ w0 (h0) ¼ w (p h0) ¼ w0 (p h0) ¼ 0 meaning that
components of the velocity vector vanish at the regular end
points of the domain. Since h ¼ 0 and h ¼ p are singular points
of the interval 0 < h p, the solution w(h) for the complete
sphere S given by Eq. (4) with the singular end points is defined
on an open interval 0 < h < p satisfying the boundary conditions
lim wðhÞ ¼ lim wðhÞ ¼ lim sin hw0 ðhÞ ¼ lim sin w0 ðhÞ ¼ 0
h!0
h!p
h!0
h!p
meaning that the components of the vorticity
@ sin hv/
@vh
@h
@/
vanish at the singular end points of the domain.
The treatment of the geometric singularity in spherical
coordinates has for many years been a difficulty in the development of analytic and numerical simulations for oceanic and
atmospheric flows around the Earth. In particular, vorticity
equations were considered by Ben-Yu28 using the spectral
method and pressure stabilization method has been developed
in Shen.11 In terms of numerical simulations, most of studies
use a powerful Jacobi-Davidson “QZ” method (Sleijpen and
Van der Vorst29) to solve the resulting linearized eigenvalue
problem (see also Weijer et al.7 for implementation of this
method in studying the barotropic Rossby basin modes of the
Argentine Basin). The common results on computational
experiments seem to provide credible evidence to support the
assertion that singular solutions to the shallow water equations
may exist on a stationary sphere. They also suggest that singular solutions are less likely on a rotating sphere. However, the
experiments conduced up to the date are not sufficiently
extensive to support any credible assertions about the existence or nonexistence of singular solutions to the shallow
water equations on a rotating sphere. Even less is known about
singular terms used on a spherical surface. In terms of analytic
treatment, the stability analysis of stationary flows in the limiting case z0 ! 1 has been considered in our earlier works in
Refs. 13 and 16. The method is based on calculating eigenvalues that are closest to a prespecified target value.
Figure 1 is used to show the exact solutions from entry
½1
[7] in Table I written in terms of the azimuthal velocities v/
½2
(the upper panel) and v/ (the lower panel) plotted on a
spherical surface at different values of time t. The value
R0 ¼ 68.4932 has been chosen for the following reason; for a
typical atmosphere, we can choose the characteristic velocity
and length scale to be c0 10 m/s, and r0 103 (see also
Table I in Ref. 7) so that, according to the definition for the
Rossby number R0, we have 1/Ro 0.0146. A small Rossby
number R0 signifies a system which is strongly affected by
Coriolis force, and a large Rossby number signifies a system
in which inertial and centrifugal forces dominate. As is seen
FIG. 3. (Color online) Comparison of the exact solutions from entry [7] in
½2
Table I for the velocity the velocity vh component plotted on a spherical
surface at the initial time t ¼ 1 and latter times of 5, 10, 15, 20, and 40.
123102-7
Nonlinear viscous fluid patterns
Phys. Fluids 23, 123102 (2011)
FIG. 4. (Color online) Comparison of the exact
solutions corresponding the entry [7] in Table I
for the pressure distribution on a spherical surface at the different values of time. The upper
panel shows the exact solution p[1] at initial
time t ¼ 1 and latter times of 5, 10, 20, and 40.
The lower panel shows the exact solution p[2] at
the same values of time.
from the analysis in this work, our model is related to the
second situation of the large Rossby numbers.
For the purpose of visualization, the maximum and minimum values of solutions in Figure 1 and the forthcoming
plots have been normalized to þ1 and 1, respectively.
Since we are only interested in qualitative analysis, the quantitative in-depth analysis including the discussion of magnitudes of the flow values for the given parameters has been
omitted from the present studies.
½1
½2
We observe the convergence of v/ and v/ as time
increases to a steady-state regime. To understand better the
nature of such convergence, we compare the latter exact solutions at certain fixed angle / versus latitude as shown in
Figure 2 at / ¼ 50 as the particular example. Namely, as we
observe from Figure 2, the convergence occurs at the vicinity
of the value v/ ¼ 100/sinh. This convergence can be
explained as follows: we remark that the general solution of
the homogeneous equation L1F ¼ 0 is given by
FðhÞ ¼
a
þ b;
sin h
(30)
where a and b are arbitrary parameters. The stationary solution (30) is the exact solution of the Navier-Stokes model
(19)–(21), a non-rotating reference frame (Ro ¼ 1) whose
asymptotic stability has been studied earlier in our work in
Ref. 16. Thus, the steady state regime shown in Figure 2 corresponds to the asymptotically stable exact solution (8) in
which we arbitrarily set a ¼ 100 and b ¼ 0.
½2
Likewise, the coalescence of the invariant solution vh
and the corresponding steady state solution can also be
observed on Figure 3 which is used to visualize the latter
invariant solution on a spherical surface. Figure 4 displays a
similar coalescence phenomena for the invariant solutions
p[1] and p[2].
Figure 5 demonstrates the spinning phenomena for the
½2
invariant solution v/ . Namely, Figure 5 shows the time se½2
ries for v/ from which we observe the rotation of atmospheric patterns around the polar axis, in an anticlockwise
sense looking above the North pole.
VI. CONCLUDING REMARKS
The exact solutions to the Navier-Stokes equations in a
thin rotating spherical shell has been found using Lie group
½2
FIG. 5. (Color online) Spinning phenomena for the invariant solution v/ on
a spherical surface (top view) at different values of time t. The atmospheric
patterns look rotating around the polar axis, in an anticlockwise sense looking above the North Pole.
FIG. 6. (Color online) Discussion of the invariant solutions w[1] and w[2]
corresponding to the entries in Table I depending on the limiting case.
123102-8
Ranis N. Ibragimov
methods. Particularly, it has been shown that the exact solutions were found as the extension of the invariant solutions for
the corresponding Euler equations. Namely, as one can check
by direct substitution, the governing Eqs. (19)–(21) are invariant with respect to the obvious translation / ¼ / þ a of the
angle /, where the constant a is the group parameter. Apparently, it is so happens that application of the approximate
equivalence transformations shows that the Navier-Stokes
equations are also invariant with respect to the one-parameter
transformation group of a more complex form, namely:
t / ¼ / þ
1 e2lR0 a ;
2R0
cos
h
2R
H
ð
h
Þ
0
w ¼ we2lR0 a þ
1 e2lR0 a :
2lR0
t ¼ te2lR0 a ;
h ¼ h;
The observation tailored to construct the exact solution presented in the entry [7] of the Table I. We also remark that
the above transformations are valid for the Euler equations
(11)–(13) which agrees with the general conclusions on the
vanishing viscosity in Cauchy’s problem for hydrodynamics
equations (Golovkin30). The summary of the exact solutions
corresponding to the limiting cases of zero/non-zero viscosity and zero/nonzero rotation is also shown schematically in
Figure 6 for different choices of F(h).
The exact solutions discussed here are used to describe
physically relevant zonal flows. For example, the presence of
the Coriolis force creates a cyclonic rotation around the poles,
i.e., west-to-east winds. From a mathematical standpoint, the
velocity and the pressure terms are unbounded in the neighborhood of the pole. Unbounded terms are common in differential equations posed in spherical coordinates and it
introduces a host of computational problems that are collectively known as the "pole problem" (see e.g., Swarztrauber19).
It has been found most recently that the nonlinear system (19)–(21) describing the dynamics of atmospheric
motion in a thin rotating shell has a remarkable property to
be self-adjoint. This property is crucial for constructing conservation laws. The more detailed group classification along
with the resulting analysis for new conservation laws for
atmospheric flows is the topic of current work and will
appear elsewhere. Additionally, an interesting topic for further studies could include an investigation of the "pole problem" in modeling of west-to-east winds from group
theoretical point of view.
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