A method is developed for computing the ground temperature accurately over both the diurnal and annual cycles. The ground is divided vertically into only two or three slabs, resulting in very efficient computation. Seasonal storage and release of heat is incorporated, and thus the method is well suited for use in climate models

Crane, G.

Lebedeff, S.

Russell, G.

English

NASA Technical Reports Server

Paper No. 59
A SLAB MODEL FOR COMPUTING GROUND TEMPERATURE IN
CLIMATE MODELS
Sergej Lebedeff, Gary Crane and Gary Russell, Goddard Institute for
Space Studies. New York. NY 10025
ABSTRACT
A method is developed for computing the ground temperature
accurately over both the diurnal and annual cycles. The ground
is divided vertically into only two or three slabs, resulting in
very efficient computation. Seasonal storage and release of
heat is incorporated, and thus the method is well suited for
use in climate models.
Introduction. The ground temperature should be computed ac-
curately over both the diurnal cycle and annual cycle in a climate model.
The diurnal and seasonal cycles are important because of the highly non-
linear way in which latent and sensible heat fluxes from the surface de-
pend upon temperature. Seasonal storage and release of heat is also im-
portant because it can amount to several watts per square meter and can
significantly influence the ground temperature (Taylor, 1976).
The basic problem is to solve the one-dimensional heat conduc-
tion equatiou
T _ 32T 3 2T
-- --- -- , (I)
8t Cg _z2 = Kg _z 2
where c is the heat capacity of the ground per unit volume, h is the
thermal g conductivity and K_ is the thermal diffusivity, subject to num-
erically specified heat flux _at the upper boundary:
F(0) = F - F, - F, - F . (2)
sw tw n q
F s is the absorbed solar radiation, F_ the net thermal radiation, and
FT_nd F are the sensible and latent he_ fluxes from the ground to the
a_mosphe%e.
A straightforward finite difference approach with computation of
temperatures at several subsurface depths can yield accurate results.
However, that method has the disadvantage of requiring (1) computation
and storage of temperatures at several depths, and (2) a short time step,
of the order of a few minutes or less.
Slab model. An approach requiring less resources, is to divide
the ground into a small number of slabs and solve for the mean tempera-
ture of each slab. Iu this method we use the average temperature of the
upper slab to simulate the surface temperature. That layer must be suf-
355
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ficiently thin that its diurnal temperature response is representative of
that for the upper skin of the ground; on the other hand, tt should be
thick enough to allow long time steps. The minimum number of ground
layers that are needed for a climate model is two, since a deep lower
layer Is needed for seasonal heat storage.
The equations for our slab model are derived under the assump-
tion that the heat capacity and conductivity are uniform In each slab, and
with the temperature profile within each slab approximated as a quad-
ratic function of depth:
2
Ti(z) = ai z +Biz +Yi" (3)
For two layers we obtain the six parameters from the six equations:
_T 1 f 0= F(0) lXI -_ 10 ' _- TldZ = TI' TI(-zl) = T2(-Zl)
-zI
-z 1 (4)
Xl _ -Zl k2"-_'- -Zl _ T2dz'¢_2' X2-- = 0.
-zl-z2 _ z ]_Zl - z2
Thus we require that no heat cross the z = -Zl-Z 2 interface.
crossing the z =-z 1 interface is:
1
3T1-3T2- 2-F(0)Zl/_l .
F(-Zl) = zl/_l + z2/_2
The heat
(5)
The final equations for the mean temperatures of the two slabs are:
ZlC 1_ = F(0) - F(-Zl) , (6)
dT2 -
z2c2 dt F(-Zl) . (7)
tion
We use a similar formulation for ocean ice but the no flux condi-
at z = -Zl-Z 2 is replaced by:
T2(-z 1- z2) = Tfo. (8)
T. , the freezing point for ocean, is taken as 271.6 K. The heat cross-
tn_the z = -z 1 interface is
12T I- IST 2 + 6Tfo - 2 F(0)ZlA 1
F(-Zl) = , (9)
4Zl/_l + 3z2/_2
356
and the heat crossing the ice-ocean interface:
T2 - Tfo F(-zQ
F(-Zl - z2) = 3_2 z2 - 2
(I0)
The final equations for the two layer model for ocean ice are:
ZlCI _ = F(0) - F(-zl) , (II)
d_2
z2c2-_ = F(-zI) - F(-zI- z2) • (12)
The equations for our three layer model are determined in a similar
way.
The thicknesses of the layers were determined empirically by
comparison with an exact periodic solution i_ud.an accurate 21 layer fin-
ite difference solution. For K_ = 4 x lo-7m _ s -l the optimum depths for
two layers were found to be zI_10 cm, z_ % 4 m, while for three layers
zl_ 5 cm, z_ 25 em, z q _ 45n. For oth%r values of K_ these depths
should be scaled in prol_rtion to the square root of K_. _ Note that for
the particular thermal dtHusivity which we employ thgdamptng depth, at
which the amplitude of the temperature variation is reduced by e-'l'_(cf.
Sellers, 1972), is_ 10 cm for the diurnal temperature wave and _2 m for
the annual temperature wave.
To illustrate the errors in the two and three layer _nodels, we
show the results of computations for the case Kg = 4 x 10- Im_s-_ (the
same as case 1 of Deardorff, 1978). For these rests we considered the
simple case of no atmosphere above the ground (Fh=Fq=0), with
Fsw = (l-Ag) So cos0 ° , (13)
FIw = e _T_ , (14)
ground albedo A = 0.24, solar constant I.354 x 103 W m -2 and ground
emissivity e = 0.e9;O isthe solar zenithangle and s the Stefan-Bolt_-
mann constant. ThiM idealizedcase yields unrealistictemperature vari-
ations at the surface, but it simplifies the computations and their inter-
pretation.
Fig. 1 compares the diurnal and annual surface temperature vari-
ations computed with several different models. The computations are
for the simplified case of no atmosphere described above. Our two layer
and three layer results are about equally accurate for the annual temper-
ature variation, as expected since both models use one deep layer for an-
nual heat storage. Three layers are clearly needed if it is desired to re-
present both the amplitude and phase of the diurnal ground temperature
variations, as well as the seasonal heat storage. We emphasize that the
Arakawa (1972) and Deardorff(1978) schemes were not designed for use in
climate models, and we have included them only to illustrate the necess-
357
25C
2a(
23(
220
20C
19¢
E= 260
E
g_
Local T_me (hours]
MnnTh
/_"'"'"' '''''"'%,,
"°
: °,
280
E
16o _
_ • L _1 t 4O
Month
Fig. I Diurnal and annual surface temperature variations computed
with several different models for t_h_ s2im_le ease of no atmosphere
and thermal dlffusivity Kg = 4 x 10- m s- . The Arakawa (1972)
and Deardorff (1978) schemes were not designed for use in climate
models, and are included here only to illustrate the necessity of
having a deep layer for annual heat storage.
ity of having a deep layer for annual heat storage. Arakawa's scheme
has one layer which is used to represent the diurnal ternperature varia-
tion. Deardorff's method has two layers, with the lower layer damping
the upper layer so as to give a realistic phase to the diurnal temperature
variation.
Fig. 2 compare s the diurnally averaged heat flux into the ground
for the same models as in Fig. 1. The Arakawa (1972) model has a ne-
gligible diurnally averaged flux, since it consists of a single thin layer
with a zero flux lower boundary condition. Although the two layers in the
Deardorff (1978) model are thin, there can be a substantial diurnally ave-
raged heat flux into (or out of) the ground because the method does not
employ a zero flux lower boundary condition.
Diseussiorz. The ground temperature model we have presented
permits efficient and accurate computation of both diurnal and seasonal
temperature variations with a minimum number of layers. It is there-
fore well suited for use in global climate models.
The diurnally averaged heat flux into the ground is usually small
in comparison to typical radiative, latent and sensible heat fluxes, even
358
_o
2° t •
x x • = I Doily Averoged Heot Flux Into Ground
x =
x I 45 N Lafitu0e,NoAtrnosphere
= •
x
x
xi
I i --_x.
_ o
,7
I1 I _N 1
_l• "'_
-20 x x•
I I I
Season (month)
Fig. 2 Diurnally averaged heat flux into the ground for the same
models as in Fig. 1. The calculations are for the simple case of
no atmosphere at 45 N latitude, as described in the text. Arakawa's
model, since it is a single thin layer with zero flux lower boundary
condition, has negligible diurnally averaged heat flux.
for this idealized ease in which the moderating effects of the atmosphere
are excluded. Thus a large impact of seasonal heat storage on short
time scales should not be expected, but the cumulative effect on season-
al time scales may be significant. Specific tests of the impact will be
included in the three-dimensional climate model described elsewhere In
these proceedings. : A previous test has been made in a general circu-
lation model by Delsol st al. (1971), who used a ground thickness of 5 m
with a lower boundary condition of T=280 K; however, they ran the model
only 14 days which is too short for an adequate test.
REFERENCES
Arakawa, A., Tech. Rep. 7_. Dept. of Meteorol., Univ. of Calif., Los
Angeles, 103, 1972.
Deardorff, J.W. , J. Geophys. Res., 83, 1889-1903, 1978.
Delsol, F., K. biiyakoda, and R.H. Clarke, Quart. J.R. Meteor. Soc.,
97, 181-208, 1971.
Sellers, W.D., Physical Climatology, Chicago, Univ. of Chicago Press,
272 pp. , 1965.
Taylor, K., J. Appl. Meteor., 15, 1129-1138, 1976.
359


A slab model for computing ground temperature in climate models

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