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Nonlinear pumping in oscillatory diffusive processes: The impact on the oceanic deep layers and lakes
Commun Nonlinear Sci Numer Simulat 19 (2014) 2131–2139
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Nonlinear pumping in oscillatory diffusive processes: The impact on the oceanic deep layers and lakes V.N. Zyryanov ⇑ Water Problems Institute, RAS, 3, Gubkina Street, Moscow 119333, Russia
a r t i c l e
i n f o
Article history: Received 25 August 2013 Received in revised form 3 October 2013 Accepted 4 October 2013 Available online 11 October 2013 Keywords: Nonlinear diffusion equation Boundary periodic problem Pumping effect Burgers equation Geophysical applications
a b s t r a c t We discuss a significant mathematical property of the nonlinear diffusion equation, socalled, the pumping effect, which of great importance in many natural diffusion processes. An oscillatory boundary value problem is considered for the nonlinear diffusive equation of thermal conductivity. We demonstrate that pure periodical oscillations of temperature at the boundary result in a nonlinear pumping of the heat implying that the heat is pumped out or into the inner regions depending on the change in the temperature oscillation amplitude. As an example, the residual effect in temperature of the World Ocean’s deep layers and lakes due to the oscillatory processes at the surface is presented and analyzed. As is generally known, the sea surface temperature (SST) profiles indicate long-term oscillations, and, therefore, according to the pumping effect when the SST oscillation amplitudes increase, the heat comes up to the surface while the deep layers become rather cooler, otherwise, as the amplitudes decrease, the heat transfers into the deep layers. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction In this paper, we consider a periodical problem for the diffusion processes that are described by the nonlinear diffusion equation,
T t ¼ ½FðTÞT z z :
ð1Þ
Many natural processes can be described by the nonlinear parabolic equations of thermal conductivity (1), in which the diffusion term FðTÞ is a function of an unknown characteristic of the medium T. Eq. (1) is of use to plenty of physical processes, e.g. the propagation of long waves toward the shelf, the fluctuations of currents in porous media, the propagation of temperature waves, polytrophic gas, etc. [1]. Although the equations of this type are referred to as thermal conductivity equations, they are often used to describe a wide range of physical processes, given that the form of the diffusion term FðTÞ varies for different classes of problems. Most frequently, FðTÞ is described by the power law FðTÞ ¼ AT n . For example, the case when n ¼ 1 emerges in plasma physics as the expansion into a vacuum of a thermalized electron cloud, as n ¼ 1 Eq. (1) is the Boussinesq equation for the dynamics of gravity filtration in a porous medium [2], at n P 1 , Eq. (1) describes gas filtration through a porous medium [3,4], at n ¼ 3 , Eq. (1) is instrumental for studying the dynamics of a thin fluid film flowing downward by gravity [5], and the evolution of long tidal gravity waves in shallow water [6–8]; and at n ¼ 6 , the equation models Marshak radiation waves [9]. A great bulk of literature has been dedicated to studying self-similar and invariant group solutions of Eq. (1). For instance, Barenblatt [10] obtained self-similar solutions of the first and second kind (incomplete self-similarity). A number of ⇑ Tel.: +7 9161595135; fax: +7 4991355415. E-mail addresses: [email protected], [email protected] 1007-5704/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2013.10.002
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remarkable solutions for the fast diffusion processes as 2 6 n 6 1 were obtained in [11]. This family of fast and superfast diffusion corresponds to processes that cease within a finite time. The works [12,13] are devoted to searching a nonclassical symmetry in the nonlinear diffusion equation. The concept of nonclassical symmetry has been introduced as a generalization of the Lie symmetry [14]. A remarkable feature of the case n ¼ 2 is that Eq. (1) can be transformed to the linear heat equa1 tion by the Blumen–Kumei contact transformation [15]. For FðTÞ ¼ ðT 2 þ 1Þ , Eq. (1) is the Mullins equation for groove development [16]. It is worth noticing that the Mullins equation, the Blumen–Kumei equation with FðTÞ ¼ T 2 , the equation with FðTÞ ¼ T 1 in plasma physics and the linear heat equation are particular cases of the Fujita equation with 1 FðTÞ ¼ ða1 þ a2 T þ a3 T 2 Þ [17]. The problem of Eq. (1) being reduced to the Fujita equation is solved in [18] allowing for certain types of the generalized conditional symmetry. Self-similar solutions are usually related to an initial problem but in nature almost all processes are periodical (or quasiperiodical). In this paper, our interest is on the periodical boundary value problem for Eq. (1). We will describe one significant effect that emerges in a periodical problem for the nonlinear thermal conduction parabolic equation. Originally this effect for quasi linear parabolic equations was described by Philip [19]. Philip derived an asymptotic solution to the steady periodic nonlinear diffusion problem. He considered the application of his theory to the Boussinesq equation to estimate the mean watertable in a coastal unconfined aquifer bounded by a straight coastline with a vertical beach connected to a sinusoidal tidal water body. He shows that the pure sinusoidal boundary condition leads to a residual effect of the mean watertable increasing at infinity. Later a similar effect of the increasing mean sea level in the problem of the tidal wave dynamics on shallows was independently obtained in works [6,7]. In [20], the authors considered the generalization of this problem to the Boussinesq equation applying to a tide-induced seawater-groundwater circulation in a multi-layered unconfined aquifer under forcing by N tidal harmonics. It is worth noting that [20] also contains a detailed review of Philip’s results. In [1], this effect was considered in relation to certain geophysical processes and was named as the pumping effect. The study showed that the pumping effect plays an important role in many physical processes. In this paper, we describe some generation of the pumping effect for Eq. (1), for the Burgers equation in particular, and discuss the possible manifestation of the pumping effect in temperature profiles of the World Ocean deep layers and lakes. The problem is that the deep waters of the North Atlantic and Arctic basins become colder despite the fact that the surface waters, on the contrary, warm up. As shown in [21,22], the deep waters of the North Atlantic are currently cooler than in the middle of the previous century. The temperature at the 1750 m depth in the North Atlantic has decreased by 0:1 to 0:4 C in comparison with the period of 1970–1974. A prominent situation has been observed in Lake Baikal [23]. The surface water temperature over the period of 1972– 2007 had been decreasing, while the temperature of the deep layers, on the contrary, had been increasing (Fig. 1). 2. Pumping effect for nonlinear parabolic equations First, we expound briefly the pumping effect according to the work [1]. Consider the one-dimensional nonlinear thermal conductivity Eq. (1). Let us look for a periodical solution of Eq. (1) within a half-line z > 0 with the following boundary conditions:
Tjz¼0 ¼ f ðtÞ;
< 1; Tjz!þ1 < C
ð2Þ
where f ðtÞ is a periodical function of period s. Introducing the averaging operator over the period,
hT i ¼
1
Z
s
tþs
T dt;
t
function WðTÞ as a primitive function of FðTÞ,
WðTÞ ¼
Z
FðTÞdT
ð3Þ
and, assuming WðTÞ is a single-valued function, the following theorem becomes true. Theorem. A periodical solution of Eq. (1) with boundary conditions (2) tends to constant T ð1Þ as z ! þ1:
T ð1Þ ¼ Wð1Þ ½hWðf ðtÞÞi;
ð4Þ
ð1Þ
where W is the inverse function of W. It is worth noting that, in the general case, T (4), one can see that hWi is an invariant along z.
ð1Þ
Proof. Taking into account the Kirchhoff substitution (3), we can rewrite Eq. (1) as
T t ¼ ½WT T z z ¼ Wzz : Averaging the left- and right-hand parts of this equation over period
s yields
does not coincide with T 0 ¼ hf ðtÞi. From
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hWizz ¼ 0 and, consequently, hWi ¼ C 1 z þ C 2 . Since hWiis the heat flux averaged over the period, it cannot grow infinitely as z ! þ1, therefore we put C 1 ¼ 0. Then hWi ¼ C 2 and hWi is an invariant independent of z. Hence, we get
hWijz¼0 ¼ hWijz!þ1
ð5Þ
As z ! þ1, oscillations attenuate and we have at infinity
hWijz!þ1 ¼ WðT ð1Þ Þ:
ð6Þ
Taking into account that
hWijz¼0 ¼ hWðf ðtÞÞi;
ð7Þ ð1Þ
and, using the inverse function W of Win (6), we obtain relation (4) from (5) and (7). Thus, a harmonic oscillation of parameter T at the domain’s boundary leads to an increase or decrease in T within the domain in comparison with the mean value at the boundary. Thus, we observe the effect of either ’pumping in’ or ’pumping out’ of substance at infinity resulted from the harmonic oscillation at the boundary. Further we will consider function f ðtÞ in (2) in the form, f ðtÞ ¼ T 0 þ T 1 cos xt , where x ¼ 2p=s. The value of invariant hWican be effortlessly obtained at infinity, where the oscillations attenuate, and, thus, Eq. (6) can be used. However, in practice, such a problem is often set over limited regions, and, then, if the problem is formulated over a limited segment 0 6 z 6 L , the procedure of finding the invariant considered in the previous section cannot however be applied at z ¼ L. In the general case, over the segment, Eq. (1) can be solved only numerically. However, if the ratio e ¼ T 1 =T 0 in the relation for f ðtÞ is infinitesimal, i.e., e << 1 , then an analytical expression for the pumping effect at the other end of the segment at z ¼ L can be obtained. Consider Eq. (1) with boundary conditions (2) and an expansion of FðT) into a series up to first-order terms with respect to e:
T t ¼ ½ða þ beT þ OðeÞÞT z z
ð8Þ
where a ¼ FðT 0 Þ, b ¼ F T ðT 0 Þ. At the right boundary, we specify the boundary condition of the second kind
T z jz¼L ¼ 0;
ð9Þ
which physically corresponds to the condition of a zero thermal flux. We look for a solution of Eq. (8) in the form of the asymptotic expansion T ¼ T 0 þ eðT ð0Þ þ eT ð1Þ þ Þ with respect to e with the boundary conditions,
T ð0Þ
z¼0
¼ T 0 cos xt;
T ð0Þ z
z¼L
¼ 0;
T ð1Þ
z¼0
¼ 0;
T zð1Þ
z¼L
¼ 0:
ð10Þ
Fig. 1. Averaged long-time temperature change in the water layers in South, Middle and North Baikal in June – September 1972–2007. [23], (a) the surface layer of 200–400 m, (b) the bottom layer (200 m from the bottom). (1) South Baikal, (2) Middle Baikal, (3) North Baikal, (4) 5-year-averaged values, (5) trend.
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We look for a solution of the first approximation T ð0Þ in the form
QðzÞeixt þ Q ðzÞeixt T ð0Þ ¼ Real Q ðzÞeixt ¼ ; 2
ð11Þ
where Real½ stands for the real part and the asterisk denotes complex conjugation. Substituting (11) into the first approximation of Eq. (8), we obtain the solution for Q ðzÞ
QðzÞ ¼ T 0
cosh½kðL zÞ ; coshðkLÞ
ð12Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where k ¼ ð1 þ iÞ x=ð2aÞ. Substituting (12) into (11) and then into the second approximation of Eq. (8) with respect to e, we obtain a solution Efor T ð1Þ , containing a periodical part and a time-independent additive, T ðÞ ðzÞ ¼ D T 0 þ eðT ð0Þ ðz; tÞ þ eT ð1Þ ðz; tÞ T 0 , that describes the pumping-effect:
T ðÞ ðzÞ ¼ e2
b ½QðzÞQ ðzÞ Qð0ÞQ ð0Þ: 4a
ð13Þ
Eq. (13) gives a quantitative value of the pumping effect at point z. If z ¼ L, the pumping effect is
T ðÞ ðLÞ ¼
bT 21 1 1 : 4a coshðkLÞ coshðk LÞ
ð14Þ
As L ! 1, we get
T ðÞ ð1Þ ¼
bT 21 : 4a
ð15Þ
According to Eq. (15), the sign of the pumping effect depends on the sign of b=a. Eqs. (12), (13) allow one to estimate the distance LðþÞ , where the mean temperature approaches the asymptotic solution of (15):
LðþÞ ¼
a 1=2 FðT Þ1=2 1 0 1 ¼ ¼ : ¼ ½2ReðkÞ kþk 2x 2x
If medium function FðTÞ in Eq. (1) is a linear function FðTÞ ¼ a þ bT (as, for example, in the cases of the temperature wave propagation in water, ice, or soil), we have the following relation for the pumping effect at infinity:
T ðÞ ¼ b
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 b þ T 21 =2; where b ¼ þ T 0 : b
ð16Þ
If b < 0, one should take the minus sign in Eq. (16), if b > 0, then the plus sign. Then, if T 1 =b << 1 and a=b >> T 0 , Eq. (16) can be simplified and reduced to Eq. (15). h
3. Numerical simulations of the pumping-effect To illustrate the pumping effect manifestation, let us consider the Fujita equation with FðTÞ in the dimensionless form
F ðT Þ ¼
ca ; b þ rT
ð17Þ
where c ¼ 10; a ¼ 2:25; b ¼ 1; r ¼ 0:596. Representing function FðTÞ in the form of (17) associated with the pumping effect applied to the ocean is further addressed in Section 5. Now, we demonstrate how the temperature amplitude change in f ðtÞ is manifested due to the pumping effect. To that end, at the surface (z ¼ 0), a periodic boundary condition T ¼ 1 þ qðt Þ sin t is applied; the heat flux at the domain bottom (z ¼ 5) is set to be zero. We define the function qðtÞ as follows: it equals 0:2 as 0 < t < 200 (the first regime); it linearly changes from 0:2 to 0:4 as 200 < t < 250; and equals 0:4 as 250 < t < 500 (the second regime). Using a function such as qðtÞ, we simulate qualitatively a situation when the temperature at the ocean surface fluctuates with a fixed amplitude up to a predefined time cutoff. After that, the fluctuation amplitude increases, for example, by a factor of two. Fig. 2 shows a numerical solution of Eq. (1) in this case. As a result, the temperature in the lower part of a basin (curve 4 in Fig. 2) reaches an asymptotic level corresponding to the first regime (level 1), and later, when the fluctuation amplitude increases, the temperature reaches another asymptotic level, corresponding to the second regime denoted as level 2. The heat loss is determined by the difference between levels 1 and 2 (interval 3). It should be stressed that as the fluctuation amplitude decreases, curve 4 rises to a higher asymptotic level. One conclusion can be drawn from the physical consideration, namely, the pumping effect is positive when FðTÞ in (1) is an increasing function, and negative when FðTÞ is a decreasing one as, for example, in Eq. (17). Indeed, the value in square brackets in Eq. (1) is the heat flux. When the temperature at the boundary periodically changes and function FðTÞ increases during the phase of greater temperatures, the heat flux into the domain is greater than the flux out of it during the phase of
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Fig. 2. Temperature T versus time at the bottom when the temperature fluctuation amplitude at the surface changes by a factor of two (numerical results, all values are dimensionless). See the text for details on curves 1–4.
the temperature decrease. As a result, the net heat flux over the period is directed into the domain, resulting in a positive pumping effect. Accordingly, a decreasing function FðTÞ results in a negative effect of the heat exchange. 4. Pumping for the Burgers equation The motion of groundwater in thin saturated layers with an inflow from both above and below, as well as flows in perforated pipes with a lateral inflow or outflow is satisfactorily described by the Burgers equation [2]. The nonlinear Burgers equation describes also diffusion processes, however, the nonlinearity enters into this equation as the advection term rather than as diffusivity. Further we will show that the pumping effect also appears in this case. Let us consider a periodic problem for the Burgers equation on the half-line
1t þ ð1x Þ2 ¼ m1xx
ð18Þ
with boundary conditions
1jx¼0 ¼ 1ð0Þ sin xt; 1jx!1 < C < 1:
ð19Þ
Eq. (18) is written in a dimensionless form, m is the dimensionless viscosity coefficient. With the help of the Hopf–Cole transformation 1 ¼ 2m lnð/Þ, Eq. (18) becomes a linear heat conductance equation over a new function /, whose solution with boundary conditions (19) has the form
/ðx; tÞ ¼
2
Z
p
1
0
1ð0Þ x2 dn: exp n2 sin x t 2m 4mn
ð20Þ
Now making use of a generating function for first kind Bessel functions to calculate the integral in (20), we obtain
/ðx; tÞ ¼
Z 1 þ1 X 2 expðinx tÞ 1ð0Þ i n x x2 pffiffiffiffi dn: Jn exp n2 2 2mi p 4mn 0 n¼1
ð21Þ
To find the limit of function /ðx; tÞ in (21) as x ! 1, we exploit the Riemann–Lebesgue lemma on integrals of oscillating functions [24]. We obtain
lim /ðx; tÞ ¼
x!þ1
2
p
J0
nð0Þ 2mi
!Z
1
2
expðn Þ dn ¼ J 0
0
! ! nð0Þ nð0Þ ¼ I0 ; 2mi 2m
ð22Þ
where I0 is the complex zero order Bessel function. Returning to the original function 1ðx; tÞ, we obtain an expression for the pumping effect for Burgers Eq. (18)
ð0Þ 1 : lim 1ðx; tÞ ¼ 1ðþÞ ¼ 2 m ln I0 x!1 2m
ð23Þ
Since I0 P 1 , the pumping effect for the Burgers equation is always positive. If the argument of function I0 is rather small, expression (23) can be simplified to the relation:
1ðþÞ
1ð0Þ 2 : 16 m
ð24Þ
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5. Estimate of the pumping effect impact on the ocean through a one-dimensional model A few past decades, a very interesting tendency is observed in the World Ocean temperature changing – the surface layers become warmer, but the deep layers yet become cooler. One of the explanations offered is that the cold penetrates into the deep layers through downwelling of the cooler surface waters in the Polar Regions or in the zones of deep convection as it occurs, for example, in the Irminger Sea [25]. However, if we were to accept this explanation, then the deep layers of the World Ocean must be warmer since the surface ocean waters as it is known have been warmer for the reason of climate warming. Thus, there must be other processes that induce the cooling of the deep layers. Moreover, simultaneously, the salinity of the deep waters in the North Atlantic has been found to increase. This also contradicts the idea of the deep layer cooling due to convection in the Polar Regions, because the ice melting results in a decrease of the surface salinity in the Arctic. The authors of the work [26] presented and analyzed data of temperature and salinity observations during 1995–2004 along the parallel of 24:50 N in the North Atlantic. One of their conclusions is that the temperature of the deep layers below 2000 m has been shown to decrease. An interesting situation has been observed in Lake Baikal [23]. The surface water temperature over the period 1972–2007 had been decreasing, while the temperature of the deep layers, on the contrary, had been increasing (Fig. 1). Obviously, this phenomenon cannot be explained by convection. Thus, one can conclude that some other processes impact on the temperature of the deep layers in seas and lakes. Further we demonstrate that it is the pumping effect that might be responsible for these phenomena. Averaging temperature T of the World Ocean over latitude and longitude, we consider a one-dimensional model of heat conductivity along the vertical direction
T t ¼ ðK ðTÞ T z Þz ;
ð25Þ
where K ðTÞ is the coefficients of the thermal conductivity. At the ocean surface, the temperature oscillates periodically about their mean values
Tjz¼0 ¼ T 0 þ T 1 cos xt; Tjz!þ1 < C 1 < 1
ð26Þ
It is of general knowledge, that the ocean thermal conductivity and salinity diffusion are governed by the processes of turbulent mixing. Since the processes of the thermal conductivity is sustained by turbulence, then it is reasonable to assume that K ðTÞ ¼ Sc K ðUÞ , where K ðUÞ is the coefficient of the vertical momentum exchange, and Sc stands for a proportionality coefficient (the Schmidt number). Below the surface Ekman layer, the parameterization of K ðUÞ , which is usual in oceanography [27,28], can be written as a function of the Brünt-Váisálá frequency NðzÞ ¼ ðg qz =q0 Þ1=2 as
K ðUÞ ðzÞ ¼ lNc ;
ð27Þ
where g is the gravity acceleration, qðzÞ is the vertical distribution of the sea water density, q0 is the mean value of the sea c water density, l ¼ d=N 1 , d ð1 2Þ 103 sm2 =s2 ; 0:5 6 c 6 2 and N 0 is the characteristic value of the Brünt-Váisálá fre0 quency. As reported in [27], the most acceptable value is c ¼ 1. To apply directly the theory of the pumping effect given in Section 2, we express K ðUÞ as a function of density instead of density gradient, i.e., in the form K ðUÞ ¼ KðqÞ. To this end, we use a hyperbolic approximation for the Brünt-Váisálá frequency below the Ekman layer in the geostrophic domain, obtained in [29],
NðzÞ ¼
hE N E ; z þ hE
ð28Þ
where hE is the upper Ekman layer thickness and N E is the value of the Brünt-Váisálá frequency at the lower boundary of the Ekman layer. Since axis z is directed downwards, z = 0 corresponds to the upper boundary of the geostrophic region in the ocean, which is the lower boundary of the Ekman layer. So, taking into account (28), we can write
N2 ðzÞ ¼
g @q ¼ q0 @ z
hE N E z þ hE
2 ð29Þ
:
Integrating (29) from bottom z ¼ H to level z, we obtain 2
2
hE N2E g h N2 ¼ ðqðHÞ qðzÞÞ þ E E : z þ hE q 0 H þ hE Substituting hE N E =ðz þ hE Þ into (28) and then into (27), we obtain
" c
c
K ðUÞ ðqÞ ¼ lhE NE
2
h N2 ðqðHÞ qðzÞÞ þ E E q0 H þ hE g
#c :
ð30Þ
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We consider the seawater state equation in a simple linear approximation form as q ¼ qH ½1 aT ðT T H Þ, where aT is the coefficient of water thermal expansion, qH ¼ qðHÞ; T H ¼ T ðHÞ are the values of the density and temperature at the bottom. So, we obtain the following equation in form (1)
T t ¼ Sc ½FðTÞT z z ;
ð31Þ
where
A ; ðB þ RTÞc
FðTÞ ¼
A¼
l hcE NcE ; B ¼
2
hE N 2E g qH aT TH; H þ hE q0
R¼
g qH aT
q0
ð32Þ
:
It is worth noticing that for the cases of integer values c ¼ 1 and c ¼ 2, Eq. (31) becomes the Fujita equation. 5.1. A homogeneous ocean model In the case of a thermally homogeneous ocean as T ðzÞ ¼ T 0 with a periodic variability of the water temperature at the surface, T ¼ T 0 þ T 1 cos x t , as c ¼ 1, we obtain the following expression for the pumping effect,
T ð1Þ
R T 21 : 4 ðB þ R T 0 Þ
ð33Þ
In the case of c ¼ 2 , we obtain the expression for the pumping effect:
T1 ¼
1 R
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðB þ R T 0 Þ2 ðR T 1 Þ2 B :
If we represent T 1 as T 1 ¼ T 0 þ T ð1Þ , then, using Eq. (34), provided the condition
T ð1Þ
R T 21 2 ðB þ R T 0 Þ
ð34Þ R T1 BþR T 0
<< 1, we can write
:
So, when c ¼ 1, the temperature pumping effect is negative with the value two times smaller than in the case of c ¼ 2. Finally, wepcan find an expression for the pumping effect in another extreme case when c ¼ 1=2. If c ¼ 1=2, the antideffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rivative is 2RA B þ R T , and we obtain the following expression:
T ð Þ 1 ¼
1 16 2 ; ð B þ R T þ R T Þ E ð k Þ ð B þ R T Þ 0 1 0 R 4 p2
2
R T1 2a where k ¼ 1þa < 1; a ¼ BþR and EðkÞ is the complete elliptic integral of the second kind. If k << 1, we obtain: T0
T ð1Þ
R T 21 : 8 ðB þ R T 0 þ R T 1 Þ
ð35Þ
Thus, for all the values of c the temperature pumping effect in the World Ocean is negative. The width of the layer, in which the temperature oscillations are observed and below which the temperature almost reaches the pumping value of T ðÞ , is equal to
LðþÞ ¼
A 2xðB þ RT 0 Þc
1=2 :
ð36Þ
This is called the Stokes layer thickness. The estimate indicates that the Stokes layer thickness, below which the temperature reaches the asymptotic level, is much smaller than the ocean total depth, therefore, to assess the pumping effect impact one can exploit the approximation of an infinitely deep ocean. 5.2. A thermally stratified ocean model The estimates of heat variation within the abyssal oceanic layers obtained in Section 5.1 correspond to the case of a thermally homogeneous ocean. This background state is described by Eq. (1), TðzÞ ¼ T 0 , as the vertical heat flux is assumed to be zero. If one considers a nonzero vertical heat flux, then another stationary solution of Eq. (1), which corresponds to a thermally stratified ocean, can be obtained. If c ¼ 1 in Eq. (31), then we obtain the stationary solution of Eq. (31):
T ðzÞ ¼
1 ½B þ C 1 expðC 2 R zÞ; R
ð37Þ
where
C 1 ¼ RT S þ B;
C2 ¼
1 RT H þ B ln ; RH RT S þ B
ð38Þ
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Fig. 3. Temperature drop at the depth of 2000 m relative to the stationary solution (37) after the annual oscillation starting of the ocean surface temperature (numerical solution).
with T S being the surface water temperature, T H being the bottom water temperature, and H being the flow depth. Solutions (37) and (38) describe a stationary exponential temperature distribution over the ocean depth with the surface temperature T S and the bottom temperature T H . Fig. 3 illustrates a numerical solution of Eq. (1) using thermal conductivity function (32) with c ¼ 1. Stationary solution (37) with T S ¼ 10 °C and T H ¼ 2:08 °C is taken as the initial conditions. At the ocean surface, the boundary condition, Tðt; zÞjz¼0 ¼ 10 þ 0:7 sin xt , with the fluctuation period of 2p=x ¼ 1 year, is specified. The temperature fluctuation amplitude at the ocean surface is 0:7 °C. Using data from the Pacific Ocean [30], we calculate coefficients B ¼ 3:27 103 m=s2 , R ¼ 1:67 103 m=s2 ð CÞ in (32). Since coefficient A does not impact on the magnitude of the pumping effect and determines only the Stokes layer thickness (36), in our calculations, we assume A ¼ 4 103 m3 =s3 . 5.3. Effect of a temperature decrease (increase) in the deep layers As one can see in Fig. 3, as periodical variations of the ocean surface temperature start, the temperature at any point of the water column decreases to a lower value. In accordance with the calculations, the temperature drop at the depth of 2000 m is 2:5 103 (°C). So, the values of the steady deep layer temperature do not correspond to steady-state solution (37). As we assessed, in the ocean, the temperature pumping effect is negative for all admissible c. In addition, our estimate does not depend on A, and as a result, it does not depend on the Schmidt number Sc. Thus, when the temperature fluctuation amplitude at the surface T 1 increases, the temperature in the ocean deep layers decreases, that is, the heat is pumped out from the depth, and inversely, when, in comparison with the previous time period, the temperature fluctuation amplitude T 1 decreases, the temperature in the ocean depth increases, that is, the heat penetrates into the deep layers. Fig. 2 depicts the results of the numerical calculations, that support our conclusion. If c ¼ 1, relation (33) becomes D T ð1Þ ¼ 1:9 102 (°C) . This is two times less than in the case of c ¼ 2, thus, all the estimates obtained above must be halved. In the case of c ¼ 1=2, from relation (35) it follows that D T ð1Þ ¼ 9 103 (°C). Now let us consider the case of Lake Baikal. Fig. 1 depicts the long-term temperature changes of the surface and bottom layers of Lake Baikal. From Fig. 1 it can be seen, that the average temperature of the lake’s surface layers over the period of 1972–2007 had decreased, that is, the surface waters had become cooler. Simultaneously, the bottom water temperature had risen up (see Fig. 1b). We believe that this temperature growth cannot be explained by convection, and suggest that the pumping effect mostly affects it. Indeed, according to Fig. 1, the long-term fluctuation amplitude of the surface water temperature during the period of 1972–2007 had decreased almost by half. Making use of the pumping-effect theory, a decrease in the surface water temperature fluctuation amplitude will lead the heat to be pumped into the deep layers, and subsequently result in an increase of the deep water temperature. It seems that it is this mechanism that acts in Lake Baikal. 6. Conclusion and discussion We presented and analyzed a significant mathematical feature, the pumping effect, of the nonlinear diffusion equation for boundary oscillatory problems. The effect has been demonstrated to impact on plenty of natural diffusion processes. We argued that pure periodical oscillations of temperature at the boundary result in a nonlinear pumping of the heat implying that the heat is pumped out or into the inner area depending on the amplitude change in the temperature oscillation. The estimates obtained in this study indicate that the pumping effect may serve as a strong nonlinear mechanism to redistribute heat in the Earth’s hydrosphere. Therefore, we draw a conclusion that a part of the heat that warms the
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atmosphere during the periods of climate warming may be transferred from the ocean deep layers. The additional heat flux from the ocean deep layers due to the pumping effect was estimated as 0:18 – 0:25 W=m2 . This assessment is comparable with the lower value of the flux due to the greenhouse effect 0:35 W=m2 . Thus, the ocean deep layers serve as a heat storage (accumulator) for the Earth given an increase of the long-term temperature fluctuation amplitude at the ocean surface. It seems that this situation has been observed during the past 50 years. Due to the negative temperature pumping effect in the ocean, the heat is pumped up from the deep ocean to the atmosphere, and on the contrary, during the period of a SST amplitude decrease, the heat penetrates into the deep layers. Our estimates indicate that, during the Earth climate warming period, the ocean deep layers can become cooler by ð0:9 – 3:6Þ 102 ð CÞ. Also, as an example, we presented applicability of the theory to Lake Baikal. During the period of 1972–2007, the longtime oscillation amplitude of the surface temperature had decreased, while the deep layer temperature of the lake had increased (Fig. 1). Such behaviour conforms completely with the pumping effect theory. Acknowledgement The reported study was partially supported by RFBR, research project: 13-05-00131. References [1] Zyryanov VN, Khublaryan MG. Pumping effect in the theory of nonlinear processes of the thermal conductivity equation type and its applications in geophysics. Dokl Earth Sci 2006;408:618–25. [2] Polubarinova-Kochina PY. Theory of groundwater movement. Princeton University Press; 1962. [3] Leibenzon L. Underground fluid dynamics: collection of papers. Moscow: Izd. AS USSR; 1952. [4] Muskat M. The flow of homogeneous fluids through porous media. N.Y.: McGraw Hill; 1937. [5] Buckmaster J. Viscous sheets advancing over dry beds. J Fluid Mech 1977;81:735–56. [6] Le Blond P. On tidal propagation in shallow rivers. J Geophys Res 1978;83:4717–21. [7] Zyryanov V, Muzylev S. Nonlinear pumping of level by tides in shallows. Dokl Acad Sci USSR 1988;298:454–8. [8] Zyryanov V. Topographic vortexes in sea current dynamics. M.: Water Problems Institute of RAS; 1995. [9] Larsen E, Pomraning G. Asymptotic analysis of nonlinear Marshak waves. SIAM J Appl Math 1980;39:201–12. [10] Barenblatt G. Similarity self similarity, intermediate asymptotics. Leningrad: Gidrometeoizdat; 1982. [11] Rosenau P. Fast and superfast diffusion processes. Phys Rev Lett 1995;7:1056–9. [12] Gandarias M, Bruzon M. Nonclassical symmetry reductions for an inhomogeneous nonlinear diffusion equation. Commun Nonlinear Sci Numer Simul 2008;13:508–16. [13] Murata S. Nonclassical symmetry of nonlinear diffusion equation and factorization method. Commun Nonlinear Sci Numer Simul 2010;15:3313–5. [14] Olver P. Applications of Lie groups to differential equations. Berlin: Springer; 1986. 2 [15] Bluman GW, Kumei S. On the remarkable nonlinear diffusion equation @=@x½aðu þ bÞ @u=@x @u=@t ¼ 0. J Math Phys 1980;21:1019–23. [16] Mullins WW. Theory of thermal grooving. J Appl Phys 1957;28:333–9. [17] Fujita H. The exact pattern of a concentration-dependent diffusion in a semi-infinite medium. Part II. Text Res J 1952;22:823–7. [18] Changzheng Q. Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry method. IMA J Appl Math 1999;62:283–302. [19] Philip J. Periodic nonlinear diffusion: an integral relation and its physical consequences. Aust J Phys 1973;26:513–9. [20] Li H, Jiao JJ. Tide-induced seawater–groundwater circulation in a multi-layered coastal leaky aquifer system. J Hydrol 2003;274:211–24. [21] Levitus S, Antonov J, Boyer T, Stephens C. Warming of the World Ocean. Science 2000;287:2225–9. [22] Dijkstra HA. Nonlinear physical oceanography. Springer; 2005. [23] Shimaraev M, Troitskaya E, Gnatovskiy RY. Temperature change of deep waters of Baikal Lake in 1972–2007 years. Geography Nature Resour 2009;3:68–76. [24] Olver F. Asymptotics and special function. N.Y., L: Academic Press; 1974. [25] Sarafanov A, Sokov A, Demidov A, Falina A. Warming and salinification of intermediate and deep waters in the Irminger Sea and Iceland Basin in 1997– 2006. Geophys Res Lett 2007;34:L23609. [26] Vargas-Yanez M, Parrilla G, Lavin A, Velez-Belchi P, Gonzalez-Pola C. Temperature and salinity increase in the eastern North Atlantic along the 24.5 N in the last ten years. Geophys Res Lett 2004;31:L06210. [27] Gargett AE. Vertical eddy diffusivity in the ocean interior. J Mar Res 1984;42:359–93. [28] Stigebrandt A. A model for the vertical circulation of the Baltic deep water. J Phys Oceanogr 1987;17:1772–85. [29] Monin AS, Neiman VG, Filyushkin BN. On the density stratification in the ocean. Dokl Acad Sci USSR 1970;191:1277–9. [30] Galerkin LI, Barash MB, Sapojnikov VV, Pasternak FA. The Pacific ocean. M.: Mysl Publications; 1982.
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Nonlinear pumping in oscillatory diffusive processes: The impact on the oceanic deep layers and lakes V.N. Zyryanov ⇑ Water Problems Institute, RAS, 3, Gubkina Street, Moscow 119333, Russia
a r t i c l e
i n f o
Article history: Received 25 August 2013 Received in revised form 3 October 2013 Accepted 4 October 2013 Available online 11 October 2013 Keywords: Nonlinear diffusion equation Boundary periodic problem Pumping effect Burgers equation Geophysical applications
a b s t r a c t We discuss a significant mathematical property of the nonlinear diffusion equation, socalled, the pumping effect, which of great importance in many natural diffusion processes. An oscillatory boundary value problem is considered for the nonlinear diffusive equation of thermal conductivity. We demonstrate that pure periodical oscillations of temperature at the boundary result in a nonlinear pumping of the heat implying that the heat is pumped out or into the inner regions depending on the change in the temperature oscillation amplitude. As an example, the residual effect in temperature of the World Ocean’s deep layers and lakes due to the oscillatory processes at the surface is presented and analyzed. As is generally known, the sea surface temperature (SST) profiles indicate long-term oscillations, and, therefore, according to the pumping effect when the SST oscillation amplitudes increase, the heat comes up to the surface while the deep layers become rather cooler, otherwise, as the amplitudes decrease, the heat transfers into the deep layers. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction In this paper, we consider a periodical problem for the diffusion processes that are described by the nonlinear diffusion equation,
T t ¼ ½FðTÞT z z :
ð1Þ
Many natural processes can be described by the nonlinear parabolic equations of thermal conductivity (1), in which the diffusion term FðTÞ is a function of an unknown characteristic of the medium T. Eq. (1) is of use to plenty of physical processes, e.g. the propagation of long waves toward the shelf, the fluctuations of currents in porous media, the propagation of temperature waves, polytrophic gas, etc. [1]. Although the equations of this type are referred to as thermal conductivity equations, they are often used to describe a wide range of physical processes, given that the form of the diffusion term FðTÞ varies for different classes of problems. Most frequently, FðTÞ is described by the power law FðTÞ ¼ AT n . For example, the case when n ¼ 1 emerges in plasma physics as the expansion into a vacuum of a thermalized electron cloud, as n ¼ 1 Eq. (1) is the Boussinesq equation for the dynamics of gravity filtration in a porous medium [2], at n P 1 , Eq. (1) describes gas filtration through a porous medium [3,4], at n ¼ 3 , Eq. (1) is instrumental for studying the dynamics of a thin fluid film flowing downward by gravity [5], and the evolution of long tidal gravity waves in shallow water [6–8]; and at n ¼ 6 , the equation models Marshak radiation waves [9]. A great bulk of literature has been dedicated to studying self-similar and invariant group solutions of Eq. (1). For instance, Barenblatt [10] obtained self-similar solutions of the first and second kind (incomplete self-similarity). A number of ⇑ Tel.: +7 9161595135; fax: +7 4991355415. E-mail addresses: [email protected], [email protected] 1007-5704/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2013.10.002
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remarkable solutions for the fast diffusion processes as 2 6 n 6 1 were obtained in [11]. This family of fast and superfast diffusion corresponds to processes that cease within a finite time. The works [12,13] are devoted to searching a nonclassical symmetry in the nonlinear diffusion equation. The concept of nonclassical symmetry has been introduced as a generalization of the Lie symmetry [14]. A remarkable feature of the case n ¼ 2 is that Eq. (1) can be transformed to the linear heat equa1 tion by the Blumen–Kumei contact transformation [15]. For FðTÞ ¼ ðT 2 þ 1Þ , Eq. (1) is the Mullins equation for groove development [16]. It is worth noticing that the Mullins equation, the Blumen–Kumei equation with FðTÞ ¼ T 2 , the equation with FðTÞ ¼ T 1 in plasma physics and the linear heat equation are particular cases of the Fujita equation with 1 FðTÞ ¼ ða1 þ a2 T þ a3 T 2 Þ [17]. The problem of Eq. (1) being reduced to the Fujita equation is solved in [18] allowing for certain types of the generalized conditional symmetry. Self-similar solutions are usually related to an initial problem but in nature almost all processes are periodical (or quasiperiodical). In this paper, our interest is on the periodical boundary value problem for Eq. (1). We will describe one significant effect that emerges in a periodical problem for the nonlinear thermal conduction parabolic equation. Originally this effect for quasi linear parabolic equations was described by Philip [19]. Philip derived an asymptotic solution to the steady periodic nonlinear diffusion problem. He considered the application of his theory to the Boussinesq equation to estimate the mean watertable in a coastal unconfined aquifer bounded by a straight coastline with a vertical beach connected to a sinusoidal tidal water body. He shows that the pure sinusoidal boundary condition leads to a residual effect of the mean watertable increasing at infinity. Later a similar effect of the increasing mean sea level in the problem of the tidal wave dynamics on shallows was independently obtained in works [6,7]. In [20], the authors considered the generalization of this problem to the Boussinesq equation applying to a tide-induced seawater-groundwater circulation in a multi-layered unconfined aquifer under forcing by N tidal harmonics. It is worth noting that [20] also contains a detailed review of Philip’s results. In [1], this effect was considered in relation to certain geophysical processes and was named as the pumping effect. The study showed that the pumping effect plays an important role in many physical processes. In this paper, we describe some generation of the pumping effect for Eq. (1), for the Burgers equation in particular, and discuss the possible manifestation of the pumping effect in temperature profiles of the World Ocean deep layers and lakes. The problem is that the deep waters of the North Atlantic and Arctic basins become colder despite the fact that the surface waters, on the contrary, warm up. As shown in [21,22], the deep waters of the North Atlantic are currently cooler than in the middle of the previous century. The temperature at the 1750 m depth in the North Atlantic has decreased by 0:1 to 0:4 C in comparison with the period of 1970–1974. A prominent situation has been observed in Lake Baikal [23]. The surface water temperature over the period of 1972– 2007 had been decreasing, while the temperature of the deep layers, on the contrary, had been increasing (Fig. 1). 2. Pumping effect for nonlinear parabolic equations First, we expound briefly the pumping effect according to the work [1]. Consider the one-dimensional nonlinear thermal conductivity Eq. (1). Let us look for a periodical solution of Eq. (1) within a half-line z > 0 with the following boundary conditions:
Tjz¼0 ¼ f ðtÞ;
< 1; Tjz!þ1 < C
ð2Þ
where f ðtÞ is a periodical function of period s. Introducing the averaging operator over the period,
hT i ¼
1
Z
s
tþs
T dt;
t
function WðTÞ as a primitive function of FðTÞ,
WðTÞ ¼
Z
FðTÞdT
ð3Þ
and, assuming WðTÞ is a single-valued function, the following theorem becomes true. Theorem. A periodical solution of Eq. (1) with boundary conditions (2) tends to constant T ð1Þ as z ! þ1:
T ð1Þ ¼ Wð1Þ ½hWðf ðtÞÞi;
ð4Þ
ð1Þ
where W is the inverse function of W. It is worth noting that, in the general case, T (4), one can see that hWi is an invariant along z.
ð1Þ
Proof. Taking into account the Kirchhoff substitution (3), we can rewrite Eq. (1) as
T t ¼ ½WT T z z ¼ Wzz : Averaging the left- and right-hand parts of this equation over period
s yields
does not coincide with T 0 ¼ hf ðtÞi. From
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hWizz ¼ 0 and, consequently, hWi ¼ C 1 z þ C 2 . Since hWiis the heat flux averaged over the period, it cannot grow infinitely as z ! þ1, therefore we put C 1 ¼ 0. Then hWi ¼ C 2 and hWi is an invariant independent of z. Hence, we get
hWijz¼0 ¼ hWijz!þ1
ð5Þ
As z ! þ1, oscillations attenuate and we have at infinity
hWijz!þ1 ¼ WðT ð1Þ Þ:
ð6Þ
Taking into account that
hWijz¼0 ¼ hWðf ðtÞÞi;
ð7Þ ð1Þ
and, using the inverse function W of Win (6), we obtain relation (4) from (5) and (7). Thus, a harmonic oscillation of parameter T at the domain’s boundary leads to an increase or decrease in T within the domain in comparison with the mean value at the boundary. Thus, we observe the effect of either ’pumping in’ or ’pumping out’ of substance at infinity resulted from the harmonic oscillation at the boundary. Further we will consider function f ðtÞ in (2) in the form, f ðtÞ ¼ T 0 þ T 1 cos xt , where x ¼ 2p=s. The value of invariant hWican be effortlessly obtained at infinity, where the oscillations attenuate, and, thus, Eq. (6) can be used. However, in practice, such a problem is often set over limited regions, and, then, if the problem is formulated over a limited segment 0 6 z 6 L , the procedure of finding the invariant considered in the previous section cannot however be applied at z ¼ L. In the general case, over the segment, Eq. (1) can be solved only numerically. However, if the ratio e ¼ T 1 =T 0 in the relation for f ðtÞ is infinitesimal, i.e., e << 1 , then an analytical expression for the pumping effect at the other end of the segment at z ¼ L can be obtained. Consider Eq. (1) with boundary conditions (2) and an expansion of FðT) into a series up to first-order terms with respect to e:
T t ¼ ½ða þ beT þ OðeÞÞT z z
ð8Þ
where a ¼ FðT 0 Þ, b ¼ F T ðT 0 Þ. At the right boundary, we specify the boundary condition of the second kind
T z jz¼L ¼ 0;
ð9Þ
which physically corresponds to the condition of a zero thermal flux. We look for a solution of Eq. (8) in the form of the asymptotic expansion T ¼ T 0 þ eðT ð0Þ þ eT ð1Þ þ Þ with respect to e with the boundary conditions,
T ð0Þ
z¼0
¼ T 0 cos xt;
T ð0Þ z
z¼L
¼ 0;
T ð1Þ
z¼0
¼ 0;
T zð1Þ
z¼L
¼ 0:
ð10Þ
Fig. 1. Averaged long-time temperature change in the water layers in South, Middle and North Baikal in June – September 1972–2007. [23], (a) the surface layer of 200–400 m, (b) the bottom layer (200 m from the bottom). (1) South Baikal, (2) Middle Baikal, (3) North Baikal, (4) 5-year-averaged values, (5) trend.
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We look for a solution of the first approximation T ð0Þ in the form
QðzÞeixt þ Q ðzÞeixt T ð0Þ ¼ Real Q ðzÞeixt ¼ ; 2
ð11Þ
where Real½ stands for the real part and the asterisk denotes complex conjugation. Substituting (11) into the first approximation of Eq. (8), we obtain the solution for Q ðzÞ
QðzÞ ¼ T 0
cosh½kðL zÞ ; coshðkLÞ
ð12Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where k ¼ ð1 þ iÞ x=ð2aÞ. Substituting (12) into (11) and then into the second approximation of Eq. (8) with respect to e, we obtain a solution Efor T ð1Þ , containing a periodical part and a time-independent additive, T ðÞ ðzÞ ¼ D T 0 þ eðT ð0Þ ðz; tÞ þ eT ð1Þ ðz; tÞ T 0 , that describes the pumping-effect:
T ðÞ ðzÞ ¼ e2
b ½QðzÞQ ðzÞ Qð0ÞQ ð0Þ: 4a
ð13Þ
Eq. (13) gives a quantitative value of the pumping effect at point z. If z ¼ L, the pumping effect is
T ðÞ ðLÞ ¼
bT 21 1 1 : 4a coshðkLÞ coshðk LÞ
ð14Þ
As L ! 1, we get
T ðÞ ð1Þ ¼
bT 21 : 4a
ð15Þ
According to Eq. (15), the sign of the pumping effect depends on the sign of b=a. Eqs. (12), (13) allow one to estimate the distance LðþÞ , where the mean temperature approaches the asymptotic solution of (15):
LðþÞ ¼
a 1=2 FðT Þ1=2 1 0 1 ¼ ¼ : ¼ ½2ReðkÞ kþk 2x 2x
If medium function FðTÞ in Eq. (1) is a linear function FðTÞ ¼ a þ bT (as, for example, in the cases of the temperature wave propagation in water, ice, or soil), we have the following relation for the pumping effect at infinity:
T ðÞ ¼ b
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 b þ T 21 =2; where b ¼ þ T 0 : b
ð16Þ
If b < 0, one should take the minus sign in Eq. (16), if b > 0, then the plus sign. Then, if T 1 =b << 1 and a=b >> T 0 , Eq. (16) can be simplified and reduced to Eq. (15). h
3. Numerical simulations of the pumping-effect To illustrate the pumping effect manifestation, let us consider the Fujita equation with FðTÞ in the dimensionless form
F ðT Þ ¼
ca ; b þ rT
ð17Þ
where c ¼ 10; a ¼ 2:25; b ¼ 1; r ¼ 0:596. Representing function FðTÞ in the form of (17) associated with the pumping effect applied to the ocean is further addressed in Section 5. Now, we demonstrate how the temperature amplitude change in f ðtÞ is manifested due to the pumping effect. To that end, at the surface (z ¼ 0), a periodic boundary condition T ¼ 1 þ qðt Þ sin t is applied; the heat flux at the domain bottom (z ¼ 5) is set to be zero. We define the function qðtÞ as follows: it equals 0:2 as 0 < t < 200 (the first regime); it linearly changes from 0:2 to 0:4 as 200 < t < 250; and equals 0:4 as 250 < t < 500 (the second regime). Using a function such as qðtÞ, we simulate qualitatively a situation when the temperature at the ocean surface fluctuates with a fixed amplitude up to a predefined time cutoff. After that, the fluctuation amplitude increases, for example, by a factor of two. Fig. 2 shows a numerical solution of Eq. (1) in this case. As a result, the temperature in the lower part of a basin (curve 4 in Fig. 2) reaches an asymptotic level corresponding to the first regime (level 1), and later, when the fluctuation amplitude increases, the temperature reaches another asymptotic level, corresponding to the second regime denoted as level 2. The heat loss is determined by the difference between levels 1 and 2 (interval 3). It should be stressed that as the fluctuation amplitude decreases, curve 4 rises to a higher asymptotic level. One conclusion can be drawn from the physical consideration, namely, the pumping effect is positive when FðTÞ in (1) is an increasing function, and negative when FðTÞ is a decreasing one as, for example, in Eq. (17). Indeed, the value in square brackets in Eq. (1) is the heat flux. When the temperature at the boundary periodically changes and function FðTÞ increases during the phase of greater temperatures, the heat flux into the domain is greater than the flux out of it during the phase of
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Fig. 2. Temperature T versus time at the bottom when the temperature fluctuation amplitude at the surface changes by a factor of two (numerical results, all values are dimensionless). See the text for details on curves 1–4.
the temperature decrease. As a result, the net heat flux over the period is directed into the domain, resulting in a positive pumping effect. Accordingly, a decreasing function FðTÞ results in a negative effect of the heat exchange. 4. Pumping for the Burgers equation The motion of groundwater in thin saturated layers with an inflow from both above and below, as well as flows in perforated pipes with a lateral inflow or outflow is satisfactorily described by the Burgers equation [2]. The nonlinear Burgers equation describes also diffusion processes, however, the nonlinearity enters into this equation as the advection term rather than as diffusivity. Further we will show that the pumping effect also appears in this case. Let us consider a periodic problem for the Burgers equation on the half-line
1t þ ð1x Þ2 ¼ m1xx
ð18Þ
with boundary conditions
1jx¼0 ¼ 1ð0Þ sin xt; 1jx!1 < C < 1:
ð19Þ
Eq. (18) is written in a dimensionless form, m is the dimensionless viscosity coefficient. With the help of the Hopf–Cole transformation 1 ¼ 2m lnð/Þ, Eq. (18) becomes a linear heat conductance equation over a new function /, whose solution with boundary conditions (19) has the form
/ðx; tÞ ¼
2
Z
p
1
0
1ð0Þ x2 dn: exp n2 sin x t 2m 4mn
ð20Þ
Now making use of a generating function for first kind Bessel functions to calculate the integral in (20), we obtain
/ðx; tÞ ¼
Z 1 þ1 X 2 expðinx tÞ 1ð0Þ i n x x2 pffiffiffiffi dn: Jn exp n2 2 2mi p 4mn 0 n¼1
ð21Þ
To find the limit of function /ðx; tÞ in (21) as x ! 1, we exploit the Riemann–Lebesgue lemma on integrals of oscillating functions [24]. We obtain
lim /ðx; tÞ ¼
x!þ1
2
p
J0
nð0Þ 2mi
!Z
1
2
expðn Þ dn ¼ J 0
0
! ! nð0Þ nð0Þ ¼ I0 ; 2mi 2m
ð22Þ
where I0 is the complex zero order Bessel function. Returning to the original function 1ðx; tÞ, we obtain an expression for the pumping effect for Burgers Eq. (18)
ð0Þ 1 : lim 1ðx; tÞ ¼ 1ðþÞ ¼ 2 m ln I0 x!1 2m
ð23Þ
Since I0 P 1 , the pumping effect for the Burgers equation is always positive. If the argument of function I0 is rather small, expression (23) can be simplified to the relation:
1ðþÞ
1ð0Þ 2 : 16 m
ð24Þ
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5. Estimate of the pumping effect impact on the ocean through a one-dimensional model A few past decades, a very interesting tendency is observed in the World Ocean temperature changing – the surface layers become warmer, but the deep layers yet become cooler. One of the explanations offered is that the cold penetrates into the deep layers through downwelling of the cooler surface waters in the Polar Regions or in the zones of deep convection as it occurs, for example, in the Irminger Sea [25]. However, if we were to accept this explanation, then the deep layers of the World Ocean must be warmer since the surface ocean waters as it is known have been warmer for the reason of climate warming. Thus, there must be other processes that induce the cooling of the deep layers. Moreover, simultaneously, the salinity of the deep waters in the North Atlantic has been found to increase. This also contradicts the idea of the deep layer cooling due to convection in the Polar Regions, because the ice melting results in a decrease of the surface salinity in the Arctic. The authors of the work [26] presented and analyzed data of temperature and salinity observations during 1995–2004 along the parallel of 24:50 N in the North Atlantic. One of their conclusions is that the temperature of the deep layers below 2000 m has been shown to decrease. An interesting situation has been observed in Lake Baikal [23]. The surface water temperature over the period 1972–2007 had been decreasing, while the temperature of the deep layers, on the contrary, had been increasing (Fig. 1). Obviously, this phenomenon cannot be explained by convection. Thus, one can conclude that some other processes impact on the temperature of the deep layers in seas and lakes. Further we demonstrate that it is the pumping effect that might be responsible for these phenomena. Averaging temperature T of the World Ocean over latitude and longitude, we consider a one-dimensional model of heat conductivity along the vertical direction
T t ¼ ðK ðTÞ T z Þz ;
ð25Þ
where K ðTÞ is the coefficients of the thermal conductivity. At the ocean surface, the temperature oscillates periodically about their mean values
Tjz¼0 ¼ T 0 þ T 1 cos xt; Tjz!þ1 < C 1 < 1
ð26Þ
It is of general knowledge, that the ocean thermal conductivity and salinity diffusion are governed by the processes of turbulent mixing. Since the processes of the thermal conductivity is sustained by turbulence, then it is reasonable to assume that K ðTÞ ¼ Sc K ðUÞ , where K ðUÞ is the coefficient of the vertical momentum exchange, and Sc stands for a proportionality coefficient (the Schmidt number). Below the surface Ekman layer, the parameterization of K ðUÞ , which is usual in oceanography [27,28], can be written as a function of the Brünt-Váisálá frequency NðzÞ ¼ ðg qz =q0 Þ1=2 as
K ðUÞ ðzÞ ¼ lNc ;
ð27Þ
where g is the gravity acceleration, qðzÞ is the vertical distribution of the sea water density, q0 is the mean value of the sea c water density, l ¼ d=N 1 , d ð1 2Þ 103 sm2 =s2 ; 0:5 6 c 6 2 and N 0 is the characteristic value of the Brünt-Váisálá fre0 quency. As reported in [27], the most acceptable value is c ¼ 1. To apply directly the theory of the pumping effect given in Section 2, we express K ðUÞ as a function of density instead of density gradient, i.e., in the form K ðUÞ ¼ KðqÞ. To this end, we use a hyperbolic approximation for the Brünt-Váisálá frequency below the Ekman layer in the geostrophic domain, obtained in [29],
NðzÞ ¼
hE N E ; z þ hE
ð28Þ
where hE is the upper Ekman layer thickness and N E is the value of the Brünt-Váisálá frequency at the lower boundary of the Ekman layer. Since axis z is directed downwards, z = 0 corresponds to the upper boundary of the geostrophic region in the ocean, which is the lower boundary of the Ekman layer. So, taking into account (28), we can write
N2 ðzÞ ¼
g @q ¼ q0 @ z
hE N E z þ hE
2 ð29Þ
:
Integrating (29) from bottom z ¼ H to level z, we obtain 2
2
hE N2E g h N2 ¼ ðqðHÞ qðzÞÞ þ E E : z þ hE q 0 H þ hE Substituting hE N E =ðz þ hE Þ into (28) and then into (27), we obtain
" c
c
K ðUÞ ðqÞ ¼ lhE NE
2
h N2 ðqðHÞ qðzÞÞ þ E E q0 H þ hE g
#c :
ð30Þ
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We consider the seawater state equation in a simple linear approximation form as q ¼ qH ½1 aT ðT T H Þ, where aT is the coefficient of water thermal expansion, qH ¼ qðHÞ; T H ¼ T ðHÞ are the values of the density and temperature at the bottom. So, we obtain the following equation in form (1)
T t ¼ Sc ½FðTÞT z z ;
ð31Þ
where
A ; ðB þ RTÞc
FðTÞ ¼
A¼
l hcE NcE ; B ¼
2
hE N 2E g qH aT TH; H þ hE q0
R¼
g qH aT
q0
ð32Þ
:
It is worth noticing that for the cases of integer values c ¼ 1 and c ¼ 2, Eq. (31) becomes the Fujita equation. 5.1. A homogeneous ocean model In the case of a thermally homogeneous ocean as T ðzÞ ¼ T 0 with a periodic variability of the water temperature at the surface, T ¼ T 0 þ T 1 cos x t , as c ¼ 1, we obtain the following expression for the pumping effect,
T ð1Þ
R T 21 : 4 ðB þ R T 0 Þ
ð33Þ
In the case of c ¼ 2 , we obtain the expression for the pumping effect:
T1 ¼
1 R
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðB þ R T 0 Þ2 ðR T 1 Þ2 B :
If we represent T 1 as T 1 ¼ T 0 þ T ð1Þ , then, using Eq. (34), provided the condition
T ð1Þ
R T 21 2 ðB þ R T 0 Þ
ð34Þ R T1 BþR T 0
<< 1, we can write
:
So, when c ¼ 1, the temperature pumping effect is negative with the value two times smaller than in the case of c ¼ 2. Finally, wepcan find an expression for the pumping effect in another extreme case when c ¼ 1=2. If c ¼ 1=2, the antideffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rivative is 2RA B þ R T , and we obtain the following expression:
T ð Þ 1 ¼
1 16 2 ; ð B þ R T þ R T Þ E ð k Þ ð B þ R T Þ 0 1 0 R 4 p2
2
R T1 2a where k ¼ 1þa < 1; a ¼ BþR and EðkÞ is the complete elliptic integral of the second kind. If k << 1, we obtain: T0
T ð1Þ
R T 21 : 8 ðB þ R T 0 þ R T 1 Þ
ð35Þ
Thus, for all the values of c the temperature pumping effect in the World Ocean is negative. The width of the layer, in which the temperature oscillations are observed and below which the temperature almost reaches the pumping value of T ðÞ , is equal to
LðþÞ ¼
A 2xðB þ RT 0 Þc
1=2 :
ð36Þ
This is called the Stokes layer thickness. The estimate indicates that the Stokes layer thickness, below which the temperature reaches the asymptotic level, is much smaller than the ocean total depth, therefore, to assess the pumping effect impact one can exploit the approximation of an infinitely deep ocean. 5.2. A thermally stratified ocean model The estimates of heat variation within the abyssal oceanic layers obtained in Section 5.1 correspond to the case of a thermally homogeneous ocean. This background state is described by Eq. (1), TðzÞ ¼ T 0 , as the vertical heat flux is assumed to be zero. If one considers a nonzero vertical heat flux, then another stationary solution of Eq. (1), which corresponds to a thermally stratified ocean, can be obtained. If c ¼ 1 in Eq. (31), then we obtain the stationary solution of Eq. (31):
T ðzÞ ¼
1 ½B þ C 1 expðC 2 R zÞ; R
ð37Þ
where
C 1 ¼ RT S þ B;
C2 ¼
1 RT H þ B ln ; RH RT S þ B
ð38Þ
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Fig. 3. Temperature drop at the depth of 2000 m relative to the stationary solution (37) after the annual oscillation starting of the ocean surface temperature (numerical solution).
with T S being the surface water temperature, T H being the bottom water temperature, and H being the flow depth. Solutions (37) and (38) describe a stationary exponential temperature distribution over the ocean depth with the surface temperature T S and the bottom temperature T H . Fig. 3 illustrates a numerical solution of Eq. (1) using thermal conductivity function (32) with c ¼ 1. Stationary solution (37) with T S ¼ 10 °C and T H ¼ 2:08 °C is taken as the initial conditions. At the ocean surface, the boundary condition, Tðt; zÞjz¼0 ¼ 10 þ 0:7 sin xt , with the fluctuation period of 2p=x ¼ 1 year, is specified. The temperature fluctuation amplitude at the ocean surface is 0:7 °C. Using data from the Pacific Ocean [30], we calculate coefficients B ¼ 3:27 103 m=s2 , R ¼ 1:67 103 m=s2 ð CÞ in (32). Since coefficient A does not impact on the magnitude of the pumping effect and determines only the Stokes layer thickness (36), in our calculations, we assume A ¼ 4 103 m3 =s3 . 5.3. Effect of a temperature decrease (increase) in the deep layers As one can see in Fig. 3, as periodical variations of the ocean surface temperature start, the temperature at any point of the water column decreases to a lower value. In accordance with the calculations, the temperature drop at the depth of 2000 m is 2:5 103 (°C). So, the values of the steady deep layer temperature do not correspond to steady-state solution (37). As we assessed, in the ocean, the temperature pumping effect is negative for all admissible c. In addition, our estimate does not depend on A, and as a result, it does not depend on the Schmidt number Sc. Thus, when the temperature fluctuation amplitude at the surface T 1 increases, the temperature in the ocean deep layers decreases, that is, the heat is pumped out from the depth, and inversely, when, in comparison with the previous time period, the temperature fluctuation amplitude T 1 decreases, the temperature in the ocean depth increases, that is, the heat penetrates into the deep layers. Fig. 2 depicts the results of the numerical calculations, that support our conclusion. If c ¼ 1, relation (33) becomes D T ð1Þ ¼ 1:9 102 (°C) . This is two times less than in the case of c ¼ 2, thus, all the estimates obtained above must be halved. In the case of c ¼ 1=2, from relation (35) it follows that D T ð1Þ ¼ 9 103 (°C). Now let us consider the case of Lake Baikal. Fig. 1 depicts the long-term temperature changes of the surface and bottom layers of Lake Baikal. From Fig. 1 it can be seen, that the average temperature of the lake’s surface layers over the period of 1972–2007 had decreased, that is, the surface waters had become cooler. Simultaneously, the bottom water temperature had risen up (see Fig. 1b). We believe that this temperature growth cannot be explained by convection, and suggest that the pumping effect mostly affects it. Indeed, according to Fig. 1, the long-term fluctuation amplitude of the surface water temperature during the period of 1972–2007 had decreased almost by half. Making use of the pumping-effect theory, a decrease in the surface water temperature fluctuation amplitude will lead the heat to be pumped into the deep layers, and subsequently result in an increase of the deep water temperature. It seems that it is this mechanism that acts in Lake Baikal. 6. Conclusion and discussion We presented and analyzed a significant mathematical feature, the pumping effect, of the nonlinear diffusion equation for boundary oscillatory problems. The effect has been demonstrated to impact on plenty of natural diffusion processes. We argued that pure periodical oscillations of temperature at the boundary result in a nonlinear pumping of the heat implying that the heat is pumped out or into the inner area depending on the amplitude change in the temperature oscillation. The estimates obtained in this study indicate that the pumping effect may serve as a strong nonlinear mechanism to redistribute heat in the Earth’s hydrosphere. Therefore, we draw a conclusion that a part of the heat that warms the
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atmosphere during the periods of climate warming may be transferred from the ocean deep layers. The additional heat flux from the ocean deep layers due to the pumping effect was estimated as 0:18 – 0:25 W=m2 . This assessment is comparable with the lower value of the flux due to the greenhouse effect 0:35 W=m2 . Thus, the ocean deep layers serve as a heat storage (accumulator) for the Earth given an increase of the long-term temperature fluctuation amplitude at the ocean surface. It seems that this situation has been observed during the past 50 years. Due to the negative temperature pumping effect in the ocean, the heat is pumped up from the deep ocean to the atmosphere, and on the contrary, during the period of a SST amplitude decrease, the heat penetrates into the deep layers. Our estimates indicate that, during the Earth climate warming period, the ocean deep layers can become cooler by ð0:9 – 3:6Þ 102 ð CÞ. Also, as an example, we presented applicability of the theory to Lake Baikal. During the period of 1972–2007, the longtime oscillation amplitude of the surface temperature had decreased, while the deep layer temperature of the lake had increased (Fig. 1). Such behaviour conforms completely with the pumping effect theory. Acknowledgement The reported study was partially supported by RFBR, research project: 13-05-00131. References [1] Zyryanov VN, Khublaryan MG. Pumping effect in the theory of nonlinear processes of the thermal conductivity equation type and its applications in geophysics. Dokl Earth Sci 2006;408:618–25. [2] Polubarinova-Kochina PY. Theory of groundwater movement. Princeton University Press; 1962. [3] Leibenzon L. Underground fluid dynamics: collection of papers. Moscow: Izd. AS USSR; 1952. [4] Muskat M. The flow of homogeneous fluids through porous media. N.Y.: McGraw Hill; 1937. [5] Buckmaster J. Viscous sheets advancing over dry beds. J Fluid Mech 1977;81:735–56. [6] Le Blond P. On tidal propagation in shallow rivers. J Geophys Res 1978;83:4717–21. [7] Zyryanov V, Muzylev S. Nonlinear pumping of level by tides in shallows. Dokl Acad Sci USSR 1988;298:454–8. [8] Zyryanov V. Topographic vortexes in sea current dynamics. M.: Water Problems Institute of RAS; 1995. [9] Larsen E, Pomraning G. Asymptotic analysis of nonlinear Marshak waves. SIAM J Appl Math 1980;39:201–12. [10] Barenblatt G. Similarity self similarity, intermediate asymptotics. Leningrad: Gidrometeoizdat; 1982. [11] Rosenau P. Fast and superfast diffusion processes. Phys Rev Lett 1995;7:1056–9. [12] Gandarias M, Bruzon M. Nonclassical symmetry reductions for an inhomogeneous nonlinear diffusion equation. Commun Nonlinear Sci Numer Simul 2008;13:508–16. [13] Murata S. Nonclassical symmetry of nonlinear diffusion equation and factorization method. Commun Nonlinear Sci Numer Simul 2010;15:3313–5. [14] Olver P. Applications of Lie groups to differential equations. Berlin: Springer; 1986. 2 [15] Bluman GW, Kumei S. On the remarkable nonlinear diffusion equation @=@x½aðu þ bÞ @u=@x @u=@t ¼ 0. J Math Phys 1980;21:1019–23. [16] Mullins WW. Theory of thermal grooving. J Appl Phys 1957;28:333–9. [17] Fujita H. The exact pattern of a concentration-dependent diffusion in a semi-infinite medium. Part II. Text Res J 1952;22:823–7. [18] Changzheng Q. Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry method. IMA J Appl Math 1999;62:283–302. [19] Philip J. Periodic nonlinear diffusion: an integral relation and its physical consequences. Aust J Phys 1973;26:513–9. [20] Li H, Jiao JJ. Tide-induced seawater–groundwater circulation in a multi-layered coastal leaky aquifer system. J Hydrol 2003;274:211–24. [21] Levitus S, Antonov J, Boyer T, Stephens C. Warming of the World Ocean. Science 2000;287:2225–9. [22] Dijkstra HA. Nonlinear physical oceanography. Springer; 2005. [23] Shimaraev M, Troitskaya E, Gnatovskiy RY. Temperature change of deep waters of Baikal Lake in 1972–2007 years. Geography Nature Resour 2009;3:68–76. [24] Olver F. Asymptotics and special function. N.Y., L: Academic Press; 1974. [25] Sarafanov A, Sokov A, Demidov A, Falina A. Warming and salinification of intermediate and deep waters in the Irminger Sea and Iceland Basin in 1997– 2006. Geophys Res Lett 2007;34:L23609. [26] Vargas-Yanez M, Parrilla G, Lavin A, Velez-Belchi P, Gonzalez-Pola C. Temperature and salinity increase in the eastern North Atlantic along the 24.5 N in the last ten years. Geophys Res Lett 2004;31:L06210. [27] Gargett AE. Vertical eddy diffusivity in the ocean interior. J Mar Res 1984;42:359–93. [28] Stigebrandt A. A model for the vertical circulation of the Baltic deep water. J Phys Oceanogr 1987;17:1772–85. [29] Monin AS, Neiman VG, Filyushkin BN. On the density stratification in the ocean. Dokl Acad Sci USSR 1970;191:1277–9. [30] Galerkin LI, Barash MB, Sapojnikov VV, Pasternak FA. The Pacific ocean. M.: Mysl Publications; 1982.