Non-linear pumping effect in oscillatory diffusive processes and its physical consequences for the World Ocean deep layers and lakes
The physical mechanism of heat transmission in oscillatory processes is described. The effect manifestation in the process of integral heat exchange in the World Ocean deep layers is studied with-in the frame of a simple one-dimensional approach. The sea surface temperature (SST) has long-term oscil...
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irk-123456789-1050962016-08-07T03:02:16Z Non-linear pumping effect in oscillatory diffusive processes and its physical consequences for the World Ocean deep layers and lakes Zyryanov, V.N. Термогидродинамика океана The physical mechanism of heat transmission in oscillatory processes is described. The effect manifestation in the process of integral heat exchange in the World Ocean deep layers is studied with-in the frame of a simple one-dimensional approach. The sea surface temperature (SST) has long-term oscillations with amplitudes greater than the trend in mean-temperature increase. The oscillations of SST lead to nonlinear pumping effect in oscillatory processes; heat is pumped out from or into the deep layers, depending on oscillation amplitudes. With increasing SST oscillation amplitudes, the heat comes out and deep layers are cooled, otherwise, with decreasing amplitudes, the heat spreads into the deep layers. Описаний фізичний механізм передачі тепла в ході коливальних процесів. У рамках простої одновимірної моделі демонструється прояв цього ефекту в процесі інтегрального теплообміну в глибинних шарах Світового океану. Спостерігаються довготривалі коливання температури поверхні моря, амплітуда яких перевищує тренд при зростанні середньої температури. Коливання поверхневої температури унаслідок нелінійності призводять до ефекту накачування: тепло викачується з глибинних шарів або поступає в них залежно від амплітуд коливань. Із збільшенням амплітуди коливань температури поверхні моря тепло виходить з глибинних шарів і вони остуджуються, а при зменшенні амплітуди тепло розповсюджується в глибинні шари. Описан физический механизм передачи тепла в ходе колебательных процессов. В рамках простой одномерной модели демонстрируется проявление этого эффекта в процессе интегрального теплообмена в глубинных слоях Мирового океана. Наблюдаются долговременные колебания температуры поверхности моря, амплитуда которых превышает тренд при возрастании средней температуры. Колебания поверхностной температуры вследствие нелинейности приводят к эффекту накачки: тепло выкачивается из глубинных слоев или поступает в них в зависимости от амплитуд колебаний. С увеличением амплитуды колебаний температуры поверхности моря тепло выходит из глубинных слое. 2013 Article Non-linear pumping effect in oscillatory diffusive processes and its physical consequences for the World Ocean deep layers and lakes / V.N. Zyryanov // Морской гидрофизический журнал. — 2013. — № 5. — С. 18-35. — Бібліогр.: 23 назв. — англ. 0233-7584 http://dspace.nbuv.gov.ua/handle/123456789/105096 551.465 en Морской гидрофизический журнал Морський гідрофізичний інститут НАН України |
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Термогидродинамика океана Термогидродинамика океана |
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Термогидродинамика океана Термогидродинамика океана Zyryanov, V.N. Non-linear pumping effect in oscillatory diffusive processes and its physical consequences for the World Ocean deep layers and lakes Морской гидрофизический журнал |
| description |
The physical mechanism of heat transmission in oscillatory processes is described. The effect manifestation in the process of integral heat exchange in the World Ocean deep layers is studied with-in the frame of a simple one-dimensional approach. The sea surface temperature (SST) has long-term oscillations with amplitudes greater than the trend in mean-temperature increase. The oscillations of SST lead to nonlinear pumping effect in oscillatory processes; heat is pumped out from or into the deep layers, depending on oscillation amplitudes. With increasing SST oscillation amplitudes, the heat comes out and deep layers are cooled, otherwise, with decreasing amplitudes, the heat spreads into the deep layers. |
| format |
Article |
| author |
Zyryanov, V.N. |
| author_facet |
Zyryanov, V.N. |
| author_sort |
Zyryanov, V.N. |
| title |
Non-linear pumping effect in oscillatory diffusive processes and its physical consequences for the World Ocean deep layers and lakes |
| title_short |
Non-linear pumping effect in oscillatory diffusive processes and its physical consequences for the World Ocean deep layers and lakes |
| title_full |
Non-linear pumping effect in oscillatory diffusive processes and its physical consequences for the World Ocean deep layers and lakes |
| title_fullStr |
Non-linear pumping effect in oscillatory diffusive processes and its physical consequences for the World Ocean deep layers and lakes |
| title_full_unstemmed |
Non-linear pumping effect in oscillatory diffusive processes and its physical consequences for the World Ocean deep layers and lakes |
| title_sort |
non-linear pumping effect in oscillatory diffusive processes and its physical consequences for the world ocean deep layers and lakes |
| publisher |
Морський гідрофізичний інститут НАН України |
| publishDate |
2013 |
| topic_facet |
Термогидродинамика океана |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/105096 |
| citation_txt |
Non-linear pumping effect in oscillatory diffusive processes and its physical consequences for the World Ocean deep layers and lakes / V.N. Zyryanov // Морской гидрофизический журнал. — 2013. — № 5. — С. 18-35. — Бібліогр.: 23 назв. — англ. |
| series |
Морской гидрофизический журнал |
| work_keys_str_mv |
AT zyryanovvn nonlinearpumpingeffectinoscillatorydiffusiveprocessesanditsphysicalconsequencesfortheworldoceandeeplayersandlakes |
| first_indexed |
2025-07-07T16:19:00Z |
| last_indexed |
2025-07-07T16:19:00Z |
| _version_ |
1837005685125021696 |
| fulltext |
© V.N. Zyryanov, 2013
UDC 551.465
V.N. Zyryanov1,2
Non-linear pumping effect in oscillatory diffusive processes and its
physical consequences for the World Ocean deep layers and lakes
The physical mechanism of heat transmission in oscillatory processes is described. The effect
manifestation in the process of integral heat exchange in the World Ocean deep layers is studied with-
in the frame of a simple one-dimensional approach. The sea surface temperature (SST) has long-term
oscillations with amplitudes greater than the trend in mean-temperature increase. The oscillations of
SST lead to nonlinear pumping effect in oscillatory processes; heat is pumped out from or into the
deep layers, depending on oscillation amplitudes. With increasing SST oscillation amplitudes, the
heat comes out and deep layers are cooled, otherwise, with decreasing amplitudes, the heat spreads
into the deep layers.
Keywords: pumping effect, heat transmission in oscillatory processes, sea surface temperature,
one-dimensional model, deep layers of the World Ocean.
1. Introduction. At the moment, there is no clear explanation for the deep wa-
ters of the North Atlantic and Arctic basin getting colder. As shown in [1, 2], deep
waters of the North Atlantic are cooler than before. The temperature at 1750 m
depth in the North Atlantic decreased by –0.1 to –0.4°C in comparison with the
period 1970 – 1974. Fig. 1 demonstrates heat content variations in ocean waters in
the North Atlantic and North Pacific in the layer of 1000 – 3000 m. One can con-
clude that the deep waters became cooler in comparison with the period 1975 –
1980. The deep-water cooling is more pronounced in the Arctic. The interannual
variability of the deep-water heat content near the Atlantic sector of the Arctic ba-
sin is 80% due to water layer change, whereas the long-term variability is caused
by the water temperature change, accounting for 60% [3]. As reported in [4], the
temperature of the Arctic basin deep waters decreased by –0.03°C from 1950 to
1978 and the maximum of the long-term water temperature decrease is –0.08°C at
1000 m in the period of observations up to 1998; at the same time, the temperature
of upper layers (< 400 m) increased (Fig. 2). An interesting phenomenon is associ-
ated with the coldest waters of the World Ocean. In 1898, F. Nansen discovered
waters with temperature of –1.3°C in the northern Norwegian Sea. Later, up to
1950, the temperature of these waters was not less than –1.1°C. However, starting
from 1970, water temperature was quickly decreasing and in 1977 achieved a value
of –1.2°C. In that period, the heat content of a 2 km deep layer decreased by more
than 40 kcal/cm2. At the same time, the average temperature of the bottom waters
in the Arctic American – Asian sub-basin also decreased. During nearly 20 years,
bottom water temperature in the Norwegian Sea decreased by –0.3°C, the maximal
temperature of the deep Atlantic waters, by –0.1°C, and the average temperature of
deep waters in the American – Asian sub-basin, by –0.05°C. The decrease in the
temperature of the Barents branch of the Atlantic waters in the Arctic, a natural
phenomenon still to be explained, is especially pronounced at the place where this
ISSN 0233-7584. Мор. гидрофиз. журн., 2013, № 5 18
branch merges with Fram's flow in the northern part of the Kara Sea. The Barents
Current core became cooler by –0.2° to –0.4°C [5].
F i g. 1. Heat content variations of the deep water layer 1000 – 3000 m for years 1950 – 1995 in the
North Atlantic (1) and the North Pacific (2) calculated by author from data of the paper [1]
F i g. 2. Distributions of temperature in the layer 200 – 1000 m in the Arctic Ocean [4]: 1 – 4 — in
1973 – 1976 years, 5 – in 1998 (the temperature at 1000 m decreased by about –0.08°C in 1998 in
comparison with 1973 – 1976, but the temperature of upper layers (< 400 m) increased)
ISSN 0233-7584. Мор. гидрофиз. журн., 2013, № 5 19
In the article [6] data of temperature and salinity observations in 1995 – 2004
on the parallel 24.5°N in the North Atlantic are described. The temperature of deep
layers below 2000 m is shown to be decreased. A widespread idea is that cold pen-
etrates into deep layers through downwelling of cooled surface waters in polar re-
gions or in zones of deep convection as, for example, in the Irminger Sea [7]. If we
proceed from this supposition, the deep layers of the World Ocean have to be
warmer because in polar regions, the surface waters become warmer. As one can
see from Fig. 2, the surface layers become warmer, but deep layers become cooler.
So, it is impossible to explain the cooling of deep layers only by water convection
in polar regions. Moreover, at the same time, the salinity of deep waters in the
North Atlantic was found to increase. This does not agree with the idea of deep
layers cooling by convection in polar regions, because ice melting leads to a de-
crease in surface salinity in the Arctic.
A more interesting situation is observed in Lake Baikal [8]. The surface water
temperature in 1972 – 2007 was decreasing, while the temperature of deep layers in
the lake was increasing (Fig. 3). Obviously, this phenomenon cannot be explained
by convection.
F i g. 3. Averaged long-time temperature change in water layers in the South, Middle and North
Baikal in June – September, 1972 – 2007 [8]: а – surface layer (200 – 400 m), b – bottom layer
(200 m from bed)
ISSN 0233-7584. Мор. гидрофиз. журн., 2013, № 5 20
From this, one can conclude that some other process can influence the temper-
ature of deep layers in seas and lakes. In this article, it is shown that vertical turbu-
lent diffusion of heat and salinity in oscillatory processes, the so-called pumping
effect, can significantly influence the temperature of deep layers in seas and lakes.
2. Pumping effect for nonlinear parabolic equations. Before considering
oceans and lakes, we expound briefly the theory of pumping effect. First this effect
was described in [9]. Authors of the paper [10] did not know about this paper and
obtained this effect independently, giving it the name of pumping effect.
Consider one-dimensional nonlinear thermal conductivity equation
( ) ,
∂
∂
∂
∂
=
∂
∂
z
TTF
zt
T (1)
where )(TF is the thermal conductivity «coefficient» (function), t – time, z is the
spatial coordinate.
We search for a periodical solution of the equation (1) on a half-line z > 0 with
the following boundary conditions:
( ) ∞<<+==
+∞→=
CTtTTtfT zz ,cos100 ω , (2)
here ( )tf is a periodical function with period ωπτ /2= , where ω is a frequency
of oscillations.
The equation (1) with boundary conditions (2) describes many physical pro-
cesses, e.g. the propagation of long waves on shallows, fluctuations of currents in
porous mediums, propagation of temperature waves, polytropic gas, etc. [10].
Introduce an operator of averaging over period τ:
∫
+
=
τ
τ
t
t
dtTT 1 (3)
and the function )(TΨ as the primitive function of )(TF :
∫=Ψ dTTFT )()( . (4)
Assume )(TΨ is a single-valued function. Denote the inverse function to Ψ as
)1(−Ψ . Then, the following theorem is true [10]: periodical solution of equation (1)
with boundary conditions (2) tends at +∞→z to a constant )(∞T :
[ ]))(()1()( tfT ΨΨ= −∞ . (5)
Note that, in the general case, )(∞T does not coincide with 0T . From (5) one can
see that Ψ is invariant along z (the proof was adduced in Appendix 1).
Thus, pure harmonic oscillation of parameter T at the domain's boundary leads
to an increase or decrease in T within the domain interior relative to the mean value
ISSN 0233-7584. Мор. гидрофиз. журн., 2013, № 5 21
at the boundary. Hence we observe an effect of either «pumping in» or «pumping
out» of the substance at infinity caused by harmonic oscillation at the boundary.
It is easy to find the value of invariant Ψ at infinity, because oscillations at-
tenuate there and equation (A1.4) can be used. However, in practice, such problem
occurs frequently for limited regions and, when the problem is formulated over a
limited segment Lz ≤≤0 , the procedure of finding the invariant considered in the
previous section cannot be repeated at Lz = . In the general case, equation (1) in a
segment can be solved only numerically. However, if the ratio 01 /TT=ε in the re-
lation for )(tf is small, i.e. 1<<ε , it is possible to find an analytical expression
for the pumping effect at the other end of the segment at Lz = (see Appendix 2).
Equations (A2.5), (A2.6) allow estimating the distance )(+L , where the mean
temperature approaches the asymptotic solution (A2.8) (see Appendix 2):
2/1
0
2/1
1
*)( 2
)(
2
)]Real(2[1
=
==
+
= −
+ ωω
αλ
λλ
TFL . (6)
If medium function F(T) in equation (1) is a linear function TTF βα +=)( (as, for
example, in the case of propagation of temperature waves in water, ice, or soil), we
have the following relation for the pumping effect at the infinity:
2/2
1
2)( TbbT +±−=± , where 0Tb +=
β
α . (7)
If b < 0, one should take the minus sign in equation (7); if b > 0, the plus sign is
taken; and if 1/1 <<bT and 0/ T>>βα , equation (7) is simplified and reduced to
equation (A2.8).
2.1. Numerical confirmation of the pumping effect. Consider the numerical
model experiment to demonstrate the manifestation of pumping effect. Take func-
tion )(TF in equation (1) in dimensionless form
( ) ( ),TrbacTF += (8)
where parameters are equal ,1,25.2,10 === bac 596.0=r . Note that the
presentation of function )(TF in the form (8) is associated with the application of
pumping effect to the ocean discussed in section 3 below. In the numerical calcula-
tions at the surface (z = 0), a periodic boundary condition ( ) )5/2sin(1 ttqT π+= is
applied, t – is nondimensional parameter, the heat flux at the domain bottom
(z = 5) is zero. We define the function )(tq as follows: it is 0.2 at 2000 << t (the
first regime); it linearly changes from 0.2 to 0.4 at 250200 << t ; and it is 0.4 at
500250 << t (the second regime). Using the function )(tq , we simulate a situation
when the temperature at the ocean surface fluctuates with fixed amplitude up to a
predefined time cutoff. After that, the fluctuation amplitude increases by a factor of
two. As a result, the temperature in the lower part of the ocean (curve 4 in Fig. 4)
reaches an asymptotic level corresponding to the first regime (level 1), and later,
when the fluctuation amplitude increases, the temperature reaches another asymp-
ISSN 0233-7584. Мор. гидрофиз. журн., 2013, № 5 22
totic 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).
F i g. 4. Behaviour of the temperature 0/TT in time t (nondimensional) at the bottom of a model
basin in the numerical experiment (see the text for explanations of 1 – 4 in section 2.1)
One may easily reach a general conclusion from physical considerations that
the pumping effect is positive when )(TF in (1) is an increasing function and neg-
ative when it is a decreasing function as, for example, the function (8). Indeed, the
value in square brackets in equation (1) is heat flux. When the temperature at the
boundary periodically changes and the function )(TF 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 temperature decrease. As a result, the net heat flux over the
period is directed into the domain, leading to a positive pumping effect. Similar
physical considerations for a decreasing function )(TF lead to a negative effect of
the heat exchange.
3. One-dimensional model for the World Ocean. The temperature T and the
salinity S of the World Ocean averaged over the latitude and longitude were de-
scribed by the following system of one-dimensional equations of nonlinear heat
conductivity and salinity diffusion in the vertical direction:
∂
∂
∂
∂
=
∂
∂
z
TK
zt
T
T ,
(9)
,
∂
∂
∂
∂
=
∂
∂
z
SK
zt
S
S
ISSN 0233-7584. Мор. гидрофиз. журн., 2013, № 5 23
where TK and SK are coefficients of thermal conductivity and salinity diffusion,
respectively, z is the vertical coordinate. At the ocean surface the temperature and
salinity oscillate periodically near their mean values
∞<<+=
+∞→= 1100 ),(cos CTtTTT zz ω ,
(10)
∞<<+=
+∞→= 2100 ),(cos CStSSS zz ω .
As known, ocean thermal conductivity and salinity diffusion are governed by the
processes of turbulent mixing. Since processes of thermal conductivity and salinity
diffusion were realized by turbulence, then it is reasonable to assume
UST KKK ~~ , where UK is the coefficient of vertical momentum exchange. Be-
low the surface of the Ekman layer, the parameterization of UK , generally accept-
ed in oceanology [11 – 13], can be written as a function of the Brünt – Váisálá fre-
quency
z
gzN
∂
∂
=
ρ
ρ0
)( as
( ) ,γµ −= NzKU (11)
where ( ) ;5.15.0;scm102...1; 2231
0 ≤≤×≈= −− γδδµ γN g – the acceleration of
gravity, ρ – water density, 0ρ is a mean value of ρ , and 0N is the characteristic
value of the Brünt – Váisálá frequency. As reported in [11], the most acceptable
value is .1=γ With weak stratification of the ocean deep layer, the value of ( )zN
can be very small; as a result, the relation (11) overestimates the values of UK . To
avoid that, in [12] the modification of relation (31) was suggested by introducing
an upper limit max
UU KK ≤ . Then the equation (11) reads:
( ) ( )max,min UU KNzK γµ −= . (12)
With the assumption max
UU KK ≤ , the relation (12) is valid over the entire oceanic
water column. To apply directly the theory of pumping effect given in section 2, we
have to express UK as a function of not density gradient, but the density itself, i.e. in
the form )(ρKKU = . For this purpose we use a hyperbolic law for the Brünt –
Váisálá frequency below the Ekman layer in a geostrophic domain, obtained in [14],
( ) ,
E
EE
hz
NhzN
+
= (13)
where Eh is the upper Ekman layer thickness and EN is the value of Brünt –
Váisálá frequency at the lower boundary of the Ekman layer. Note that the expo-
nential Emery – Lee – Magaard approximation [15] or the exponential Munk –
Wunsch approximation for a coefficient of a thermal conductivity [16] could be
used instead of the hyperbolic approximation of Brünt – Váisálá frequency (13).
However, comparison has shown the parameterization (13) to be in better agree-
ISSN 0233-7584. Мор. гидрофиз. журн., 2013, № 5 24
ment with the real distribution of Brünt – Váisálá frequency in the ocean. Moreo-
ver, the relationship (13) is preferable from the mathematical viewpoint.
So, taking into account (13), we can write
2
0
2 )(
+
=
∂
∂
=
E
EE
hz
Nh
z
gzN ρ
ρ
. (14)
A standard approximation for the Brünt – Váisálá frequency was made in (14) by
substituting the mean value of 0ρ instead of the density ρ in the denominator.
Integrating (14) from the bottom Hz = to the level z, we find
E
EE
E
EE
hH
NhzHg
hz
Nh
+
+−=
+
22
0
22
)]()([ ρρ
ρ
, (15)
where H is the ocean bottom. Substituting )/( EEE hzNh + from (15) in (13) and
then in (11), we obtain
( )
( )[ ]
γ
γγ
ρρ
ρ
µ
ρ
+
++−
=
E
EE
EE
U
hH
NhHzg
NhK
22
0
)(
. (16)
We take the equation of the seawater state in the simple linear approximation as
( )[ ],)(1 HSHTH SSTT −+−−= βαρρ where Tα is a coefficient of thermal expan-
sion of water, Sβ is a coefficient of salinity compression, ( ),HH ρρ = ( )HTTH = ,
( )HSSH = . Multiplying the first equation of system (9) by Tα− , and the second
equation by Sβ and adding them, we obtain the equation for the density ),( ztρ :
,)(Sc
∂
∂
∂
∂
=
∂
∂
z
K
zt
ρρρ (17)
where
γρ
ρ
)(
)(
RB
AK
−
= ;
00
22
;)(;
ρρ
ρµ γγ gRHg
hH
NhBNhA
E
EE
EE =−
+
== , (18)
Sc is the Schmidt number, i.e. the ratio of the characteristic values of the coeffi-
cient of turbulent thermal conductivity to the kinematic coefficient of turbulent
momentum exchange.
Substituting (10) into the equation of seawater state, we obtain boundary con-
ditions for water density at ocean surface
∞<<+=
+∞→= 3100 ),(cos Ct zz ρωρρρ , (19)
where
[ ])()(1 00 HSHTH SSTT −+−−= βαρρ , )( 111 TS TSH αβρρ −= . (20)
ISSN 0233-7584. Мор. гидрофиз. журн., 2013, № 5 25
In the case of a lake, 0=Sβ and we have the equation for the thermal conduc-
tivity
,)(~Sc
∂
∂
∂
∂
=
∂
∂
z
TTF
zt
T (21)
where
γ)~~(
~
)(~
TRB
ATF
+
= ;
00
22 ~;~;~
ρ
αρ
ρ
αρ
µ γγ TH
H
TH
E
EE
EE
gRTg
hH
NhBNhA =−
+
== . (22)
3.1. Homogeneous ocean. First, consider a case of a homogeneous ocean
( ) .0ρρ =z But the water density at the surface varies periodically
tωρρρ cos10 += near a value 0ρ . The lower boundary of the surface Ekman lay-
er (i.e. beginning of the geostrophic domain) is assumed to coincide with the ocean
surface. The relation (12) allows us to avoid singularity in (17) for the real ocean,
so we can consider .0>− ρRB An antiderivative function for )(ρK in (17) for
1≠γ is the function
( ) ( )
,
1 1−−−
− γργ RBR
A and for 1=γ the function
)(ln)( ρRBRA −− . At 1=γ we get the following expression for the value of the
pumping effect:
( )
( )0
2
1
4 ρ
ρ
ρ
RB
R
−
≈+
∞ . (23)
Thus, one can see from (23) that if 1=γ the pumping effect is positive. Calcula-
tions for 2/1=γ and 2=γ give the positive pumping effect for the density too.
So, increasing amplitude of surface density oscillations leads to an increase in wa-
ter density in the World Ocean deep layers.
If we assume const)( =zS then for the temperature we obtain the equation
(21) with decreasing function )(TF . Therefore the pumping effect for temperature
is negative. Similarly, one can find that the pumping effect for salinity is positive.
So, the positive pumping effect for density corresponds to negative pumping effect
for temperature and positive pumping effect for salinity.
Further we will consider the pumping effect for temperature alone. For the
thermally homogeneous ocean ( ) 0TzT = with periodic variability of water tem-
perature at the surface tTTT ωcos10 += with 1=γ , we get the following expres-
sion for the value of pumping effect
( )
( ) .4 0
2
1
TRB
TRT
+
−≈−
∞ (24)
So, for 1=γ the pumping effect is negative for temperature and its value is two
times smaller than for 2=γ . Finally, we can find an expression for the pumping
effect for the other extreme case with 2/1=γ (Appendix 3). Thus, for all values γ
the pumping effect for temperature in the World Ocean is negative.
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The layer, where the temperature oscillations are observed and after it the tem-
perature almost reaches the pumping value ( )±T , is
2/1
0
)( )(2
+
=+ γω RTB
AL . (25)
This is the Stokes layer thickness. Assuming the Schmidt number 1Sс ≈ , we get
337 s/m102 −×≈A . From (25) one can obtain that Stokes layer thickness is about
83 m for oscillations with a period of one year and about 187 m for a period of five
years. These estimates show that the thickness of Stokes layer, below which the
temperature reaches the asymptotic level, is much less than the total thickness of
the ocean; therefore, the approximation of an infinitely deep ocean is quite applica-
ble to evaluating the pumping effect.
3.2. Effect of temperature decrease (increase) in the deep layers. As we see,
the pumping effect in the World Ocean is negative for temperature for all admissi-
ble values of γ , while for density and salinity, it is positive. In addition, the value
does not depend on ;A hence, it does not depend on the Schmidt number Sc. Thus,
when the amplitude of temperature fluctuations at surface 1T increases, the tem-
perature in the depths of the ocean decreases, that is, the heat is pumped out from
the depths, and inversely, when the temperature fluctuations amplitude 1T decreas-
es as compared to the previous time period, the temperature in the ocean depths
increases, i.e. heat spreads toward deeper layers. The result of the numerical model
experiment shown in Fig. 4 illustrates this conclusion.
To estimate the amplitude increase of long-term fluctuations in the mean sur-
face temperature of the World Ocean, we use the results of [17]. Following Reid,
the relation for the globally averaged surface temperature of the World Ocean sT
and envelope curve of the solar activity fluctuations (Wolf number WN ) becomes
( )
,
4
1089.0
0s Q
NTT Wα−
+= (26)
where )Cm/(W2.2 2 °⋅=Q and 3.0=α is the total albedo of the Earth. Analysis
of fluctuations of the Solar activity (Wolf numbers) shows that, starting from 1925
– 1930, the solar activity amplitude fluctuation has been increasing. This is the up-
going phase of the 80 – 90 yrs Gleissberg period. The maximal value of the enve-
lope curve of Wolf numbers during period 1900 – 1950 is 90 in average, while the
same maximal value over 1950 – 2000 is about 190. Thus, the difference consti-
tutes about 100. We suppose that the increase in long-term temperature fluctuations
at the surface of the World Ocean is proportional to the increase in globally aver-
aged surface temperature.
Substituting this value in (26), we get an estimate of the swing amplitude in-
crease of long-term temperature fluctuations at the ocean surface °≈∆ 7.0sT C.
The amplitudes of the temperature fluctuations at the ocean surface for these peri-
ods are equal to half of the value assessed.
ISSN 0233-7584. Мор. гидрофиз. журн., 2013, № 5 27
Another estimate of fluctuations of the mean surface temperature of the World
Ocean can be derived from observational data. SST fluctuations in the equatorial
zone of the Atlantic from 1950 is shown in Fig. 5 [18]. The graph is based on data
of instrumental observations involving satellite imaginary. The amplitude increase
of long-term SST fluctuations from 1960 is clearly seen in Fig. 5. By 2000, this
increase reaches 1.5°C, i.e. it is two times greater than the estimate obtained from
(26). So, an integral estimate of °≈∆ 7.0sT C seems reasonable. Moreover, Fig. 5
confirms the hypothesis that the long oscillation amplitude of the ocean surface
temperature in 1950 – 2000 yrs increases by a factor of 1.5. We estimate the tem-
perature decrease in the deep layers in case of a homogeneous ocean when the am-
plitude of long-term temperature fluctuations at surface exceeds °≈ 35.01T C.
F i g. 5. Long-term temperature fluctuations T at the ocean surface in the equatorial zone of the At-
lantic at the point ( °° 0;S10 ) [18] (the value of minTTT −=∆ is plotted along the vertical axis, where
minT is the minimal temperature at the given point): 1 – data of observations; 2 – nonlinear trend from
the paper [18]
Let us evaluate some applied parameter values. In the World Ocean,
,km5,s10,m100 12 ≈≈≈ −− HNh EE and ( ) .C1067.1 14 −− °×=Tα We assume the
bottom temperature for the Pacific to be °≈ 08.2HT C. Then, we get
23 sm1027.3~ −×−≈B and ( )Cs /m1067.1~ 23 °⋅×≈ −R . If we also assume that the
ISSN 0233-7584. Мор. гидрофиз. журн., 2013, № 5 28
mean deep water temperature in the Pacific, °≈ 66.30T C [19], is the mean-deep
water temperature in the World Ocean, we get: .sm108.2~~ 23
0
−×≈+ TRB
Using relation (A3.1) in case of ,2=γ we get the following estimate for the
temperature decrease in the deep ocean layers: ( ) °×−=∆ −−
∞ (106.3 2T C). Now we
calculate the heat losses of the deep ocean layers over the period of climate warm-
ing. The estimated volume of the Pacific is 6101.707 × km3 the temperature de-
crease is °036.0 C. The water thermal capacity is ( )Cgcal928.0 °⋅ . These values
correspond to the heat of 2221 104.2036.0928.0101.707 ×=××× cal transported
from the depths of the Pacific.
Since the volume of the Pacific is approximately 50% of the World Ocean, our
estimate was doubled. Thus, over the period of climate warming, the heat loss of
the World Ocean deep layers owing to the pumping effect at 2=γ is about
22108.4 × cal or 23100.2 × J. Taking into account the heat transportation from the
ocean during a period of 50 – 70 years and that the Earth's surface area is
14105× m2, we obtain the specific heat flux of 25.018.0 −≈ W/m2. This value is
more than two times greater than the geothermal heat flux from the Earth's interior
(0.09 W/m2).
If 1=γ the relation (24) reads: ( ) )С(109.1 2 °×−=∆ −−
∞T . This is two times less
than in the case of 2=γ , thus, in this case, all estimates obtained above have to be
halved. In case of 2/1=γ , calculations from the relation (A3.4) lead to
( ) )С(109 3 °×−=∆ −−
∞T .
4. Thermally non-homogeneous ocean. Estimates of the heat loss given
above were made for a thermally homogeneous ocean. Such state corresponds to
one of stationary solutions of equation (1), where the thermal conductivity coeffi-
cient is in the form (22) and the vertical heat flux is zero. If we admit that the heat
flux is not zero, we can obtain a steady-state solution. For the case of 1=γ in (21),
we obtain a time-independent solution of equation (21):
( ) [ ])~~exp(~~
~
1
21 zRCCB
R
zT +−= , (27)
where
,~~
~~
ln~
1~,~~~
s
2s1 BTR
BTR
HR
CBTRC H
+
+
=+= (28)
sT is the surface water temperature, HT is its bottom temperature, and H is the
depth of the fluid.
The solution (27), (28) describes a stationary exponential temperature distribu-
tion over ocean depth with surface temperature sT and bottom temperature HT .
Fig. 6 gives a numerical solution of equation (1) with thermal conductivity function
(22) at 1=γ . The stationary solution (27) with °=10sT C and °= 08.2HT C is tak-
ISSN 0233-7584. Мор. гидрофиз. журн., 2013, № 5 29
en as the initial condition. On the ocean surface, the boundary condition
tztT z ωsin7.010),( 0 +=
=
with fluctuation period of 1/2 =ωπ year is used. The
function of thermal conductivity has the form of (22) with 31027.3~ −×−=B m/s2,
31067.1~ −×=R m/(s2·°C). Since coefficient A has no influence on the magnitude of
the pumping effect and determines only the Stokes layer thickness, in these calcu-
lations we assumed 3104~ −×=A m3/s3. The amplitude of temperature fluctuations
on the ocean surface was 0.7 according to Fig. 5. As one can see in Fig. 6, after the
start of periodic variations in ocean surface temperature, the temperature in any
point of water column goes down to low values. In accordance with this calcula-
tions, the temperature drop at the depth of 2000 m is 3105.2 −× (°C).
F i g. 6. Temperature drop at the depth 2000 m relative to the stationary solution (equation (19))
after the start of the annual oscillation of ocean surface temperature (numerical solution)
Now let us consider the case of Lake Baikal. In Fig. 3 long-term changes of
temperature of surface and bottom layers in Baikal are shown. From Fig. 3, а it can
be seen, that the average temperature of surface layers of the lake over period 1972
– 2007 decreased, i.e. surface waters were cooled. At the same time, the tempera-
ture of bottom water layers rose (Fig. 3, b). It is impossible to explain the rise of
bottom water temperature by the convection process. The only explanation that can
be offered for this phenomenon is based on the pumping effect. Indeed, as can be
ISSN 0233-7584. Мор. гидрофиз. журн., 2013, № 5 30
seen from Fig. 3, а, the amplitude of long-term fluctuations of surface water tem-
perature in period 1972 – 2007 almost halved. Under the pumping effect theory, a
decrease in the amplitude of surface water temperature fluctuations in the lake will
lead to pumping in of heat in deep layers, i.e. to an increase in the temperature of
deep waters.
5. Conclusions. The estimates obtained in this study show that the pumping
effect can serve as a strong nonlinear mechanism in processes of heat flux redistri-
bution on the Earth. As follows from the results described above, a part of the heat
that warmed the atmosphere during the periods of climate warming is transported
from the deep layers of the ocean. According to different estimates, the magnitude
of greenhouse effect is equal to the increase in additional heat flux to the Earth sur-
face from 0.35 to 4 – 5 W/m2. The additional heat flux from the deep layers of the
World Ocean as a result of the pumping effect is estimated at about 0.18 –
0.25 W/m2. It is comparable with the lower value of the flux caused by greenhouse
effect. If we assume even higher values of SST long-term fluctuations [18], the
estimates of the pumping effect magnitude become higher also.
Thus, the deep layers of the ocean serve as heat storage (accumulator) of the
Earth when the amplitude of long-term temperature fluctuations at the ocean sur-
face increases. This situation is observed during the past 50 years. Due to the nega-
tive pumping effect for temperature in the ocean, the heat is pumped up from the
deep ocean to the atmosphere; on the contrary, during period of SST amplitude
decrease, heat penetrates into the deep layers. A hypothesis explains the increase in
SST fluctuations amplitude by the existence of solar activity envelope curve [20].
With this hypothesis, one can explain the appearance of 1950 – 1970 cooling peri-
od [21]. During this period, the solar activity decreased. Following to pumping ef-
fect theory, a decrease in SST fluctuations amplitude leads to pumping-down of
heat from ocean surface to the deep ocean during the period of 1950 – 1970 and,
consequently, to a slight climate cooling.
As a result, the nonlinear pumping effect is a cause of heat exchange between
the ocean and the atmosphere. In climate-warming period, the amplitude of SST os-
cillations increases, causing cooling of the deep layers. This idea was discussed in
paper [22]. Our estimates show that, for the Earth climate warming period, the deep
layers of the World Ocean may become cooler by (0.9 – 3.6) × 10-2 (°C). Other mani-
festations of pumping effect in geophysics were considered in the work [23].
Lake Baikal demonstrates another manifestation of the pumping effect. During
the period 1972 – 2007, the amplitude of long-time oscillations of surface tempera-
ture decreased and the temperature of deep layers in the lake increased as it should
be according the pumping effect theory.
Acknowledgements This study was supported by the Russian Foundation for
Basic Research (project no. 13-05-00131).
ISSN 0233-7584. Мор. гидрофиз. журн., 2013, № 5 31
Appendix 1
Taking into account equation (4), we can rewrite equation (1) as
2
2
zz
T
dT
d
zt
T
∂
Ψ∂
≡
∂
∂Ψ
∂
∂
=
∂
∂ . (A1.1)
Averaging the left- and right-hand parts of equation (A1.1) over period τ yields
02
2
=
Ψ
zd
d
(A1.2)
and, consequently, 21 CzC +=Ψ . Since Ψ is nothing but heat flux averaged
over the period, Ψ cannot grow infinitely at +∞→z , therefore 01 =C . It fol-
lows from this that 2C=Ψ and Ψ is an invariant independent of z . As a re-
sult, we get
+∞→=
Ψ=Ψ zz 0
. (A1.3)
At +∞→z , oscillations attenuate and we have at infinity
)( )(∞
+∞→
Ψ=Ψ Tz . (A1.4)
Taking into account that
))((
0
tf
z
Ψ=Ψ
=
(A1.5)
and using the inverse function )1(−Ψ to Ψ in (A1.4), we obtain relation (5) from
(A1.3) and (A1.5).
Appendix 2
Consider equation (1) with boundary conditions (2) and an expansion of F(T) into a
series to the terms of the first order with respect to ε :
∂
∂
++
∂
∂
=
∂
∂
z
TOT
zt
T )]([ εεβα , (A2.1)
where
dT
TdFTF )(,)( 0
0 == βα . At the right end of the segment, we specify the
boundary condition of the second kind
0=
∂
∂
=Lzz
T
, (A2.2)
which physically corresponds to the condition of zero thermal flux. We search for
the solution of equation (A2.1) in the form of asymptotic expansion
...)1()0( ++= TTT ε with respect to ε with boundary conditions
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0,0,0,cos
)1(
0
)1(
)0(
0
)0( =
∂
∂
==
∂
∂
=
=
=
=
=
Lz
z
Lz
z z
TT
z
TtAT ω . (A2.3)
We search for the solution for the first approximation )0(T as
[ ]
2
e)(e)(e)(Real
*
)0(
titi
ti zQzQzQT
ωω
ω
−+
== , (A2.4)
where Real denotes the real part and the asterisk denotes complex conjugated
function. Substituting (A2.4) into the first approximation of equation (A2.1), we
obtain the solution for )(zQ :
)cosh(
)](cosh[)( 1 L
zLTzQ
λ
λ −
= , (A2.5)
where )2/()1( αωλ i+= . Substituting (A2.5) into (A2.4) and then into the second
approximation of equation (A2.1) with respect to ε , we obtain a solution for )1(T ,
containing a periodical part and a time-independent additive, which describes the
pumping effect:
[ ])0()0()()(
4
)( **)( QQzQzQzT −−=±
α
β . (A2.6)
Equation (A2.6) gives a quantitative value of the pumping effect at point z. At the
end of segment Lz = , the pumping effect will be
−−=± 1
)cosh()cosh(
1
4
)( *
2
1)(
LL
TLT
λλα
β . (A2.7)
At ∞→L , we get
α
β
4
)(
2
1)( TT =∞± . (A2.8)
As can be seen from (A2.8), the sign of pumping effect depends on the sign of
αβ / .
Appendix 3
In the case of 2=γ we get the expression for pumping effect
( ) ( )
−−+=∞ BTRTRB
R
T ~~~~
~
1 2
1
2
0 . (A3.1)
If we represent ∞T as ( )±
∞∞ += TTT 0 , then, using (A3.1), we can write with the
condition ( ) 1~~~
01 <<+ TRBTR :
( )
( ).~~2
~
0
2
1
TRB
TRT
+
−≈−
∞ (A3.2)
ISSN 0233-7584. Мор. гидрофиз. журн., 2013, № 5 33
The case 2/1=γ is the most labor-intensive for numerical calculations. In this case, an
antiderivative function is the function .~~
~
~2 TRB
R
A
+ We get the following expression:
( ) ( ) ( )
+−++=±
∞ 0
2
102
)( ~~~~~
4
16
~
1 TRBkETRTRB
R
T
π
, (A3.3)
where ( ) ( )01
2 ~~~,112 TRBTRaaak +=<+= and ( )kE is a full elliptic integral of
the second type.
Raising the elliptic integral ( )kE to the second power and leaving only the
terms up to the fourth order of magnitude respectively to ,k we get:
( )
( ).~~~8
~
10
2
1
TRTRB
TRT
++
−≈−
∞ (A3.4)
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1 Water Problems Institute, RAS, Received July 11, 2012
Moscow, Russia
E-mail: zyryanov@aqua.laser.ru
2 Lomonosov Moscow State University, Physical Department
АНОТАЦІЯ Описаний фізичний механізм передачі тепла в ході коливальних процесів. У
рамках простої одновимірної моделі демонструється прояв цього ефекту в процесі інтеграль-
ного теплообміну в глибинних шарах Світового океану. Спостерігаються довготривалі колива-
ння температури поверхні моря, амплітуда яких перевищує тренд при зростанні середньої
температури. Коливання поверхневої температури унаслідок нелінійності призводять до ефек-
ту накачування: тепло викачується з глибинних шарів або поступає в них залежно від амплітуд
коливань. Із збільшенням амплітуди коливань температури поверхні моря тепло виходить з
глибинних шарів і вони остуджуються, а при зменшенні амплітуди тепло розповсюджується в
глибинні шари.
Ключові слова: ефект накачування, передача тепла в коливальних процесах, температура
поверхні моря, одновимірна модель, глибинні шари Світового океану.
АННОТАЦИЯ Описан физический механизм передачи тепла в ходе колебательных процессов.
В рамках простой одномерной модели демонстрируется проявление этого эффекта в процессе
интегрального теплообмена в глубинных слоях Мирового океана. Наблюдаются долговремен-
ные колебания температуры поверхности моря, амплитуда которых превышает тренд при
возрастании средней температуры. Колебания поверхностной температуры вследствие нели-
нейности приводят к эффекту накачки: тепло выкачивается из глубинных слоев или поступает
в них в зависимости от амплитуд колебаний. С увеличением амплитуды колебаний темпе-
ратуры поверхности моря тепло выходит из глубинных слоев и они остужаются, а при
уменьшении амплитуды тепло распространяется в глубинные слои.
Ключевые слова: эффект накачки, передача тепла в колебательных процессах, температу-
ра поверхности моря, одномерная модель, глубинные слои Мирового океана.
ISSN 0233-7584. Мор. гидрофиз. журн., 2013, № 5 35
mailto:zyryanov@aqua.laser.ru
2. Pumping effect for nonlinear parabolic equations. Before considering oceans and lakes, we expound briefly the theory of pumping effect. First this effect was described in [9]. Authors of the paper [10] did not know about this paper and obtained thi...
Consider one-dimensional nonlinear thermal conductivity equation
3. One-dimensional model for the World Ocean. The temperature and the salinity of the World Ocean averaged over the latitude and longitude were described by the following system of one-dimensional equations of nonlinear heat conductivity and salinit...
3.2. Effect of temperature decrease (increase) in the deep layers. As we see, the pumping effect in the World Ocean is negative for temperature for all admissible values of , while for density and salinity, it is positive. In addition, the value does...
4. Thermally non-homogeneous ocean. Estimates of the heat loss given above were made for a thermally homogeneous ocean. Such state corresponds to one of stationary solutions of equation (1), where the thermal conductivity coefficient is in the form (2...
5. Conclusions. The estimates obtained in this study show that the pumping effect can serve as a strong nonlinear mechanism in processes of heat flux redistribution on the Earth. As follows from the results described above, a part of the heat that war...
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