A skin-layer parameterization for use in a turbulent kinetic energy
approach to determining sea surface temperature
Chia-Ying Tu and Ben-Jei Tsuang *
Dept. of Environmental Engineering, Nat’l Chung-Hsing University, Taichung, Taiwan 40227
*Corresponding author: tel: +886-4-22851206, fax: +886-4-22862587, email:
tsuang@nchu.edu.tw
Abstract
We presented a Skin-Layer Parameterization (SLP) and showed how it can be used in a
Turbulence Kinetic Energy (TKE) type ocean model to determine Skin Sea Surface
Temperature (SST) and to account for the warm-layer effect as well as the surface-cooling
effect. Determination of temperatures at the skin layer and at an uppermost layer with 1-mm
thickness is found sufficient to reproduce the cool-skin effect. A vertical resolution of 1-2 m
is needed to keep track of the absorption of solar radiation in the top few meters and to
reproduce the warm-layer effect. Data from research vessel Vickers taken during the TOGA
COARE program are used for a case study, in which the observed cool-skin effect and the
warm-layer effect are reproduced using the above discretizations. Our results show that a cool
skin constantly increases atmospheric heat input to the ocean by ~ 8 W m-2, and that a warm
layer decreases it by ~ 1 W m-2.
Keywords: air-sea interaction, skin-layer parameterization, conduction layer, cool skin, warm
layer, sea surface temperature, turbulent kinetic energy
1 Introduction
Sea surface temperature (SST) plays a crucial role in determining air-sea fluxes (e.g.,
Ineson and Gordon, 1989; Fairall et al., 1996), air-lake fluxes (e.g., Hostetler and Bartlein,
1990; Tsuang and Dracup, 1991) and sea ice formation. For climate studies (Gates, 1992;
1
Hostetler et al., 1993; Tsuang et al., 2001), and more generally for studies focusing on
ocean-atmosphere (lake-atmosphere) interactions, a better parameterization to determine SST
is required.
In ocean general circulation models (OGCM) there are two approaches to simulate the
thermocline of a water body: a bulk mixed layer approach and an eddy turbulent kinetic
energy approach (TKE) (Haidvogel and Bryan, 1992). Most bulk mixed layer approaches
assume the SST to be the average temperature of a mixed layer, and conventional eddy
kinetic energy approaches assume the SST to be the temperature determined in the uppermost
layer.
The bulk mixed layer approach (Niller, 1975; Davis et al., 1981; Garwood, 1977;
Garwood, 1985a; 1985b) is simple and computationally efficient, and has been used and
tuned into global ocean models (Wells, 1979; Admamec et al., 1981; Schopf and Cane, 1983;
Oberhuber, 1993a; 1993b). However, few studies verify its ability to simulate the diurnal
variation of SST.
A TKE approach (Mellor and Yamada, 1982), although more computational intensive,
determines a more realistic diurnal variation of SST (Gaspar et al., 1990; Hostetler et al.,
1993; Chia and Wu, 1998). A question always arises whether SST is determined as the
temperature in the uppermost numerical layer in TKE approaches (Martin, 1985; Gaspar et al.,
1990; Hostetler et al., 1993). If we assume that they are equal, the so-called SST thus
2
calculated is vertical resolution dependent. Gaspar et al. (1990) tested three vertical
resolutions: 0.2, 0.5 and 5 m. They found that a 0.5-m vertical resolution can capture the
diurnal variation of SST, but a 5-m resolution appeared to be insufficient to resolve the
strongly surface trapped response of an upper ocean and the large diurnal cycles of the SST
are poorly simulated. The maximum amplitude of a 5-m resolution is reduced to a few tenths
of a degree, while the observation is around one degree. As a result, a high vertical resolution
by using traditional TKE approaches is needed to simulate the diurnal variation of SST.
Hostetler et al. (1993) discretized the entire water column of lakes into 1-m thickness, Gaspar
et al. (1990) suggested a vertical resolution of 1 m from the surface down to 30 m of ocean
study sites, and Chia and Wu (1998) used a vertical resolution of 0.25 m from the surface
down to 1 m. Hence, the resulting TKE approaches are much more time-and-memory
consuming than bulk mixed layer approaches, although TKE approaches are more naturally
embedded into primitive equation solvers (Gaspar et al., 1990).
Besides, a cool-skin phenomenon complicates the problem further. It is well recognized
that the temperature at the sea skin surface is typically a few tenths degree Celsius cooler than
the temperature some tens of centimeters below (Saunders, 1967; Paulson and Simpson, 1981;
Dalu and Purini, 1982; Wu, 1985; Coppin et al., 1991; Fairall et al., 1996; Soloviev and
Schlussel, 1994; 1996; Wick et al., 1996; Donlon and Robinson, 1997; Donlon et al., 1999;
2002). Most of the temperature difference across the upper meter of ocean usually occurs in
3
the uppermost one millimeter where molecular transfer dominates. The layer is referred as a
viscous layer. Together with the viscous layer, there exists a conduction layer with a
somewhat smaller thickness of 0.2 - 0.6 mm (Khundzhua et al., 1977; Paulson and Simpson,
1981; Donlon et al., 2002), where heat transport is only possible by molecular conduction
(Hasse, 1971; Grassl, 1976). This layer is also called the cool skin of the ocean.
Fairall et al. (1996) showed that Bulk Sea Surface Temperature (BSST, temperature of
the upper few meters) data must be corrected for cool-skin and diurnal warm-layer effects to
obtain bulk surface flux estimates approaching the 10-W m-2 accuracy. A cool skin increases
the average atmospheric heat input to the ocean by ~ 10 W m-2 and a warm layer decreases it
by ~ 4 W m-2. Paulson and Simpson (1981) made a similar conclusion. A renewal theory
(Paulson and Simpson, 1981; Fairall et al., 1996; Soloviev and Schlussel, 1994; Fairall et al.,
1996; Wick et al., 1996; Donlon et al., 2002; Castro et al., 2003; Horrocks et al., 2003) has
been applied to simulate the cool-skin effect induced by molecular diffusion. But to the
author’s knowledge the theory has not been incorporated into an ocean model. Alternatively,
this study presents a numerical skin-layer parameterization (SLP) to simulate the cool-skin
effect and the diurnal warm-layer effect for a TKE approach.
Introduction of the SLP is presented in the next section. The concept of SLP has been
introduced for determining land surface skin temperature (Arakawa and Mintz, 1974;
Deardorff, 1978; Tsuang and Yuan, 1994; Tsuang and Tu, 2002; Tsuang, 2003). This study
4
extends the works of Arakawa and Mintz (1974), Tsuang and Yuan (1994) and Tsuang (2003)
to a more general formulation to account for the wavelength-dependent absorption of solar
radiation in the upper few meters and in the conduction layer. The parameterization can be
embedded into TKE approaches by solving temperatures at many levels implicitly.
Field campaign data measured from the research vessel Vickers (R/V Vickers) during the
Coupled Ocean-Atmosphere Response Experiment (COARE) of the Tropical Ocean Global
Atmosphere (TOGA) program (Webster and Lukas, 1992) is used for a case study.
2 Skin-layer parameterization (SLP)
An ideal surface is considered as shown in Figure 1. T0 is the skin temperature and T1 is
the average temperature of the uppermost layer of the water column with a thickness of h1.
The parameterization of the skin temperature is as follows. First, it is assumed that the heat
transfer under the surface can be described by the heat diffusion equation without considering
the absorption of solar radiation at this moment as:
∂T
∂ 2T
=k 2
∂t
∂z
(1)
and heat diffusion coefficient k is a constant. In addition, the skin temperature of the surface
can be described by a cosine function with the highest temperature at time tm, which can be
written as:
T0 = T0 + ∆T0 cos(ω (t − t m ))
(2)
5
where T0 is average skin temperature (K), ∆T0 is half the diurnal skin temperature
variation (K), t is local time (s) and ω is the angular velocity of the earth with respect to the
sun (=2π/86400 s-1). According to Carslaw and Jaeger (1959), the analytical solution of the
temperature profile of the aforementioned ideal case is (Pielke, 1984):
(
T (z , t ) = T0 + ∆T0 exp
(ω / 2k )z )cos(ω (t − t m ) + (ω / 2k )z )
(3)
Now let’s compare the energy budget in the region from the surface to the depth d.
Integrating Eq. (1), we have
∫
−d
0
−d
∂T
∂ 2T
dz = ∫ k 2 dz
0
∂t
∂z
G − G01
= 0
ρ w0 c w
(4)
where G0 and G01 are heat flux as shown in Figure 1 with positive downward (W m-2 ), ρ w0
and cw are reference density (kg m-3) and specific heat of water (J kg-1 K-1). Substituting Eq.
(3) into the left-hand side term of the above equation, it can be rewritten as:
6
∫
−d
0
∂T ( z , t )
dz
∂t
⎛ ω ⎞ ⎛
−d
ω ⎞⎟
z ⎟⎟ sin ⎜⎜ ω (t − t m ) +
z ⎟dz
= ∫ ω∆T0 exp⎜⎜
0
k
k
2
2
⎠
⎠ ⎝
⎝
⎧
⎤⎫
⎞ ⎡
⎛
⎟ ⎢
⎜
⎪
⎥⎪
π⎤
k⎪ ⎡
⎜ − d ⎟ cos ⎢ω (t − t ) − d + π ⎥ ⎪
(
)
ω
= −ω∆T0
−
+
t
t
cos
exp
−
⎬
⎨
m
m
⎜
ω ⎪ ⎢⎣
4 ⎦⎥
ω ⎟ ⎢
ω 4 ⎥⎪
⎟
⎜
⎥⎪
⎪⎩
2k ⎠ ⎢⎣
2k
⎦⎭
⎝
⎡
⎞⎤
⎛
⎟⎥
⎜
⎢
π⎤
k⎢
d ⎟⎥ ⎡
⎜
1 − exp −
cos ⎢ω (t − t m ) + ⎥
≈ −ω∆T0
⎟
⎜
ω⎢
4⎦
ω ⎥ ⎣
⎟⎥
⎜
⎢
2k ⎠ ⎦
⎝
⎣
= ω∆T0
⎡
⎞⎤
⎛
⎟⎥
⎜
3π ⎤
⎡
⎢
k⎢
d ⎟⎥ ⎛⎜ i ⎢⎣ω (t −tm )− 4 ⎥⎦ ⎞⎟
⎜
1 − exp −
Re e
⎜
⎟
ω⎢
ω ⎟⎥ ⎜⎝
⎠
⎟⎥
⎜
⎢
2k ⎠ ⎦
⎝
⎣
(5)
Note that, according to Eq. (2),
[
]
∂T0 ∂
= T0 + ∆T0 cos(ω (t − t m ))
∂t
∂t
= −ω∆T0 sin(ω (t − t m ))
(6)
⎛ i ⎡⎢ω (t − tm )+ π2 ⎤⎥ ⎞
⎦ ⎟
= ω∆T0 Re⎜ e ⎣
⎟
⎜
⎠
⎝
Neglecting the phase change terms and combining the above two equations to eliminate ∆T0 ,
we get
⎡
⎛
⎞⎤
⎜
⎟⎥
⎢
− d ∂T ( z , t )
k⎢
d ⎟⎥ ∂T0
⎜
∫0 ∂t dz ≈ ω ⎢1 − exp⎜ − ω ⎟⎥ ∂t
⎜
⎟⎥
⎢
2k ⎠ ⎦
⎝
⎣
∂T
= he 0
∂t
where we define the effective heat thickness he (m) of a surface with depth d as
7
(7)
⎛
⎛
⎞⎞
⎜
⎜
⎟⎟
k⎜
d ⎟⎟
⎜
he ≡
1 − exp −
⎜
ω⎜
ω ⎟⎟
⎜⎜
⎜
⎟ ⎟⎟
k
2
⎝
⎠⎠
⎝
Note that he is less than or equal to both
(8)
k / ω and d / 2 . In addition, the effective
thickness he approaches infinitely thin as the heat diffusivity approaches zero.
By combining Eq. (4) and (7), we have
∂T0
∂ 2T
= k 20
∂t
∂z
G − G01
≈ 0
ρc p he
(9)
Now, let’s modify Eq. (1) to account for the absorption of solar radiation as (Martin,
1985; Gaspar et al., 1990):
R ∂F
∂T
∂ 2T
= k 2 + sn
∂t
∂z
ρ w 0 c w ∂z
(10)
where Rsn is net solar radiation at the surface (W m-2); F(z) is fraction (dimensionless) of Rsn
that penetrates to the depth z. F(z) is parameterized according to a nine-band equation by
Soloviev and Schlussel (1996) as:
n
F ( z ) = ∑ f i exp(( z − z 0 ) / ξ i )
(11)
i =1
where z 0 is the height of water surface (m), n is the number of wavelength bands (n = 9), fi
are the fractions of solar energy in each band, and ξ i are the corresponding attenuation
lengths. Note that Soloviev and Schlussel (1996) modified the Paulson and Simpson’s (1981)
equation for clear water for various types of water defined by Jerlov (1976).
8
To account for the absorption of shortwave solar radiation absorption, according to (11)
the net heat flux absorbed within the sea surface numerical layer of depth d should be
modified as G0 + Rsn (F ( z0 ) − F ( z0 − d )) − G01 . Hence, Eq. (9) becomes:
∂T0 G0 + Rsn (F ( z 0 ) − F ( z 0 − d )) − G01
≈
∂t
ρc p he
G0
R (F ( z 0 ) − F ( z 0 − d ))
T −T
=
+ sn
−k 0 1
ρ w0 cw he
ρ w0 cw he
he ( z 0 − z1 )
(12)
where z0 and z1 are the heights (m) of T0 and T1, respectively. Note that G01 in the above
equation is determined according to a finite difference scheme as:
G01 = ρ w0 c w k
T0 − T1
z 0 − z1
(13)
Note that the first term on the right-hand side of (12) is a cooling term since the non-solar
surface heat flux G0 is usually upward (Saunders, 1967; Dalu and Purini, 1982; Soloviev and
Schlussel, 1996; Fairall et al., 1996). The second term is a heating term due to the absorption
of shortwave solar radiation (Fairall et al., 1996). The surface non-solar heat flux G0 (W m-2)
is given by:
G0 = Rld − Rlu − H − LE
(14)
It is the sum of sensible heat flux (H), latent heat flux (LE), terrestrial longwave radiation (Rlu)
and atmospheric longwave radiation (Rld). If he is small, the decrease in surface temperature
due to the negative value of G0 becomes apparent. Eq. (12) is proposed by this study to
calculate the skin temperature T0.
9
In addition, T1, the mean temperature of the upper first layer, although, can be
determined by a conventional finite difference on Eq. (10) as:
⎡
⎛ z + z ⎞⎤
Rsn ⎢ F (z0 ) − F ⎜ 1 2 ⎟⎥
∂T1 G0 − G12
⎝ 2 ⎠⎦
⎣
=
+
∂t
ρ w0 c w h1
ρ w0 c w h1
⎡
⎛ z + z ⎞⎤
G0 + Rsn ⎢ F (z 0 ) − F ⎜ 1 2 ⎟⎥
T −T
⎝ 2 ⎠⎦
⎣
=
−k 1 2
ρ w 0 c w h1
h1 (z1 − z 2 )
,
(15)
we calculate it by combining Eq. (12) and (15) to eliminate G0 term as:
Rsn ⎡
∂T1 he ∂T0
⎛ z + z ⎞⎤
F ( z 0 − d ) − F ⎜ 1 2 ⎟⎥
=
+
⎢
∂t h1 ∂t ρ w0 cw h1 ⎣
⎝ 2 ⎠⎦
T −T
T −T
+k 0 1 −k 1 2
h1 ( z 0 − z1 )
h1 ( z1 − z 2 )
(16)
The above equation can determine T1 without G0 term. This equation preserves the matrix of
the temperatures as a diagonal matrix. See Eq. (A1) - Eq. (A3) in the Appendix for the matrix
form. Note that since T1 represents the mean temperature of the top numerical layer, it keeps
track of the energy content of the layer. In contrast, the skin layer, with skin temperature T0, is
designed as a transparent medium that does not keep track of the energy content but should
mimic the skin temperature as precisely as possible. In summary, Eq. (12) and Eq. (16) are
proposed by this study to calculate the skin temperature and the mean temperature of the
uppermost layer, respectively.
3 TKE Model
This study determines the vertical profiles of temperature, momentum and salinity of a
water column as (Mellor and Durbin, 1975; Martin, 1985; Gaspar et al., 1990):
10
Rsn ∂F ∂T ' w'
∂T
=
−
∂t
ρ w0 c w ∂z
∂z
(17)
∂U
∂ U ' w'
= − fk × U −
∂t
∂z
(18)
∂S
∂ S ' w'
=−
∂t
∂z
(19)
where T, U, S, and w are temperature (K), horizontal velocity (m s-1), salinity (practical
salinity ‰), and vertical velocity, respectively; f is Coriolis parameter (s-1); and k is the
vertical unit vector. The over bar represents a time averaged value. The surface fluxes are
specified as follows:
T ' w'(0) = −G0 / ρ w 0 cw = (Rlu − Rld + H + LE ) / ρ w 0 cw
(20)
S ' w'(0 ) = −(E vap − P − R )S1
(21)
U' w'(0) = −τ / ρ w0
(22)
where Evap, P, and R are evaporation, precipitation rate and river inflow rates, respectively (m
s-1); S1 is the average salinity of the uppermost layer; and τ is surface wind shear (N m-2).
The vertical fluxes are parameterized using the classical K approach as:
T ' w' = −(k h + ν h )∂T / ∂z
(23)
U ' w' = −(k m + ν m )∂U / ∂z
(24)
S' w' = −(k h + ν h )∂ S / ∂z
(25)
where kh is eddy diffusion coefficient (m2 s-1) for heat and salinity, km is eddy diffusion
coefficient (m2 s-1) for momentum, ν h and ν m are molecular diffusion coefficients (so
called “ambient diffusion coefficient” or “ambient diffusivity”) for heat and momentum,
11
respectively (m2 s-1).
The eddy diffusion coefficient for heat within the conduction layer and the eddy
diffusion coefficient for momentum within the viscous layer are set to zero. That is,
molecular transport is the only mechanism for the vertical diffusion of heat and momentum in
the conduction layer and in the viscous layer, respectively (Hasse, 1971; Grassl, 1976; Wu,
1985). These phenomena have been confirmed in the field experiment (Khundzhua et al.,
1977). Setting a thickness of 0.5 mm for the conduction layer and setting a thickness of 1 mm
for the viscous layer (Soloviev and Schlussel, 1996; Donlon et al., 2002) are adopted in this
study. An analysis of the sensitivity of the cool-skin phenomenon to the thickness of the
conduction layer will be discussed later in the paper. Below the conduction layer and the
viscous layer, eddy diffusivity is determined according to a TKE-mixing length approach.
Note that eddy diffusivity is a function of turbulent kinetic energy (TKE) and turbulent
lengths (Mellor and Durbin, 1975; Martin, 1985; Gaspar et al., 1990). The relationship
between km and kh is determined according to the Prandtl number (Mellor and Yamada, 1982;
Martin, 1985; Gaspar et al., 1990; Pinazo et al., 1996; Leredde et al., 1999) as:
Prt =
km
kh
(26)
where Prt is turbulent Prandtl number. A unity value for Prt is used in this study. The unity
value is used in many recent numerical models (Gaspar et al., 1990; Pinazo et al., 1996;
Leredde et al., 1999) and has been reported in several field data for ocean (Gregg et al., 1985;
12
Peters et al., 1988) although Mellor and Yamada (1982) and Martin (1985) used a value of 0.8.
The eddy diffusivity for momentum km is simulated by an eddy kinetic energy approach based
on the Prandtl-Kolmogorov hypothesis as:
k m = ck l k E
(27)
where ck = 0.1 (Gaspar et al., 1990), lk is a mixing length, and E is turbulent kinetic energy,
(
)
E = 0.5 u '2 + v '2 + w '2 . The turbulent kinetic energy is determined by a one-dimensional
equation as (Mellor and Yamada, 1982):
2
3/ 2
⎛ ∂U ⎞
∂E ∂
∂E
g ∂ ρw
E
⎟ + kh
= km
+ k m ⎜⎜
c
−
ε
⎟
∂t ∂z
∂z
ρ w0 ∂z
lε
⎝ ∂z ⎠
(28)
where cε = 0.7 (Bougeault and Lacarrere, 1989; Gaspar et al., 1990), ρ w is water density,
and lε is a dissipation length. A numerical form of the above Eq. can be found in Eq. (A4)
in the Appendix. Water density is determined according to UNESCO (1981) of the standard
seawater. Gaspar et al. further imposed that the TKE never falls below a minimal value E min
of 1×10-6 m2 s-2 as a result of intermittent turbulent mixing and internal wave breaking in the
Pynocline. Note that a minimum value E min of 1×10-6 m2 s-2 is equivalent to having a
minimum value of diffusivity (nm and nh) according to Eq. (27). Nonetheless, estimates of the
ambient diffusion coefficient vary over a wide range from the small molecular values for
momentum, heat and salt (1×10-6, 1.4×10-7, 2×10-8 m2 s-1, respectively) to values of 1×10-4 to
2×10-4 m2 s-1 (Garrett, 1977). The values of E min , nm and nh are site dependent (Paulson and
Simpson, 1981; Martin, 1985). In this study E min , nm and nh are set to 1.0×10-6 m2 s-2(Gaspar
13
et al., 1990), 1.20×10-6 m2 s-1 and 1.34×10-7 m2 s-1 (Paulson and Simpson, 1981; Chia and Wu,
1998; Mellor and Durbin, 1975), respectively.
Following Gaspar et al (1990) and Pinazo et al. (1996), the mixing and dissipation
lengths were functions of two length scales which represented upward and downward
conversion of TKE (Bougeault and Lacarrere, 1989) as:
lε = l u l d
(29)
l k = min(l u , l d )
(30)
g
ρ w0
g
ρ w0
∫
z + lu
∫
z −ld
z
z
ρ w ( z ) − ρ w ( z ')dz ' = E ( z )
(31)
ρ w ( z ) − ρ w ( z ')dz ' = E ( z )
(32)
The acceptable values of z + lu and z − ld are naturally limited by the ocean surface and
bottom. That is, mixing lengths l k at the surface and the bottom according to (30) approach
zero. As a result, zero values of eddy diffusion coefficients at the surface and the bottom can
be derived according to Eq. (27). A zero value of eddy diffusivity at the ocean surface is
consistent with the assumptions made in the surface conduction layer (Grassl, 1976).
4 Case Study
Because the model we used here is one-dimensional, we needed to test it in an oceanic
region where advective effects are known to be weak. One-dimensional models (Anderson et
al., 1996; Chia and Wu, 1998) have been successfully applied in regions around the Improved
METeorological (IMET) mooring site (1.45oS, 156oE) during the TOGA experimental period
14
in the western Pacific Ocean. In addition, during the period from 1 February 1993 to 21
February 1993, a research vessel, R/V Vickers, stayed in the vicinity of 2oS and 156oE, where
standard meteorological parameters as well as solar radiation, atmospheric radiation, SST and
temperature near the surface were measured. SST was measured with a Heiman KT-19
radiometer, and was calibrated with a moving seawater bath as described by Schluessel et al.
(1990). Temperature near the surface was measured with platinum resistance thermometers at
a depth of approximately 3 m (T3m), at the cooling water intake port of the research vessel.
Downward shortwave and longwave radiations were measured with a Kipp and Zonen
Pyranometer and an Eppley Pyrgeometer, respectively, and the resulting data were used to
derive the model. Fairall et al. (1996), Soloviev and Schlussel (1996), Wick et al. (1996),
Kley et al. (1997), and Schanz and Schlussel (1997) have all used R/V Vickers’ data set for
studying cool-skin effect and warm-layer effect. More detailed descriptions of the
measurements from R/V Vickers can be found in Soloviev and Schlussel (1996) and in Wick
et al. (1996).
Terrestrial longwave radiation used in the model was calculated according to the
Stefan-Boltzman law based on simulated SST with an emissivity of 0.97 (Brutsaert, 1982).
Latent and sensible heat fluxes were determined according to the Businger equations (1971)
using the observed surface meteorological data from R/V Vickers and the simulated SST.
Surface roughness was determined according to Charnock (1955). The absorption coefficients
15
of solar radiation in Eq. (11) are set according to Fairall et al. (1996).
The model was verified by simulating temperature profiles taken at the study site. This
simulation used a fine vertical discretization W01T01S27 as listed in Table 1. W01T01S27
has ten 100-µm-thick layers in the top 1 mm, 27 numerical layers within the top 1-m depth
and a vertical resolution of 1 m within a 1 – 9 m depth. Initial water temperature and salinity
profiles, as well as precipitation data, are taken from the IMET mooring measurements.
During the simulation, salinities at all depths, and temperatures at depths greater than 10 m
were updated every hour according to the mooring observations. Updating temperatures at
depths greater than 10 m is done to reduce uncertainty about the influence of heat sources
from the water below 10 m. A time step of 900 s is used in the model.
Results
Figure 2 compares the simulated temperatures with the field measurements. It can be
seen that the diurnal patterns and tendencies of SST, T3m and ∆T3m (=SST-T3m) were well
reproduced by the model. The correlation coefficient of SST between simulations and
observations is 0.79 with a root-mean-square error (RMSE) of 0.26 K (Table 2). Figure 3
illustrates the relationship of ∆T3m as a function of solar irradiance and wind speed during
nighttime and daytime. It clearly shows that the cool-skin effect (negative ∆T3m) was well
simulated by the model under both strong and weak solar irradiance. Nonetheless, during
strong wind nights the cool-skin effect was overestimated, and during weak wind afternoons
16
the warm-layer effect (positive ∆T3m) was overestimated. As a consequence, SST were
underestimated during strong wind nights, and overestimated during weak wind afternoons.
Figure 4a illustrates composite diurnal temperature profiles from the surface down to 10
m in log scale. The figure shows that the coolest profile occurred at ~ 6 AM just before
sunrise, and the warmest was slightly phase delayed at ~ 3 PM in the afternoon. It shows that
the cool skin resided at depths within the conduction layer (0– 0.5 mm). Note that there are
five numerical layers in the conduction layer. Heat loss by means of latent heat flux and net
longwave radiation cooled the water surface, and caused the skin temperature to be colder
than that below the conduction layer by a magnitude of 0.13 K both day and night. In the
conduction layer, only molecular diffusion remains. Such a low diffusivity prohibits the
mixing of the cool surface with its underneath water and forms the cool skin. Figure 5
analyzes the sensitivity of the strength of the cool skin by varying the thickness of the
conduction layer. It shows that the strength of the former increases with the thickness of the
latter. A thickness of 0.5 mm has the lowest bias and the lowest RMSE of ∆T3m during the
nighttime periods, compared to other thicknesses. The 0.5-mm value falls within the range of
0.2– 0.6 mm measured by Khundzhua et al. (1972), and is the same as those suggested by
Soloviev and Schlussel (1996) and Donlon et al. (2002).
The warm layer was present from ~ 6 AM after sunrise until ~ 6 PM before sunset.
Figure 4a also shows that a warm layer resided at depths between 1 mm – 1 m where most
17
solar radiation was absorbed by the water. At 3 PM, within this depth, the temperature was
higher than SST by 0.28 K and higher than the 10-m temperature by 0.57 K.
5 Discussion
Although applying the above “W01T01S27” discretization has successfully captured the
warm-layer effect as well as the cool-skin effect, simulations involving 27 numerical layers
within the top 1-m depth are computationally expensive. A more economical discretization
may be done by beginning with a visual inspection of the vertical temperature profile
simulated by using the W01T01S27 discretization (Figure 4a). This can give a general idea of
the depths of the cool skin and the warm layer. It is clear that the cool skin resided at depths
within the conduction layer (0– 0.5 mm), and that the warm layer resided at depths between 1
mm – 1 m. Hence, temperatures calculated at the surface and at the bottom of the conduction
layer (0.5-mm depth) are required in order to reproduce the magnitude of the cool-skin effect,
and temperatures calculated at depths between 1 mm – 1 m are required in order to reproduce
the magnitude of the warm-layer effect. Seven coarser discretizations listed in Table 1 are
discussed with the aim of finding alternatives to compromising between the demands of
computational resources and the required accuracy. Table 2 summarizes the statistics of the
simulated temperatures of each discretization. Figure 4 shows the composite diurnal
temperature profiles using the discretizations W01T10S02, W05T10S02 and W01T50S02.
The discretizations use name conventions such as “W$$T##S%%”. “W$$” is the
18
vertical resolution in meters in the warm layer (at depths between 1 m and 10 m); “T##” is
the thickness of the top numerical layer given in units of 0.1 mm, where T01 denotes 0.1 mm
and T10 denotes 1.0 mm; “S” indicates that SLP is implemented (if SLP is not implemented,
“S” is replaced by ”N”), and “%%” represents the number of numerical layers within the top
1-m depth. If “T##S%%” is not specified, it indicates that no numerical layers exist within
the top 1 m.
Figure 6 compares the conventional discretizations W01, W02 and W05 with the
suggested discretization by adding two numerical layers: one skin layer and an uppermost
numerical layer with a thickness of 1 mm, and they are named as W01T10S02, W02T10S02
and W05T10S02, respectively. It can be seen that none of the conventional discretizations can
capture the cool-skin phenomenon. Besides, W01 and W02 generate artificial warm-layer
effects during the first half of the simulation period, and W05 is unable to reproduce the
warm-layer effect. On the other hand, all the suggested discretizations (W01T10S02,
W02T10S02 and W05T10S02) reproduce the cool skin behavior, while W01T10S02 and
W02T10S02 both reproduce the warm-layer effect better than W01 and W02.
5.1
Cool Skin Discretization
As can be seen in Figure 4d, when we used the discretization W01T50S02, where the
uppermost numerical heat flux was determined at 1.25 mm, the cool-skin effect was not
reproduced. This depth is below the conduction layer. Below the conduction layer, vertical
19
mixing is governed by eddy diffusion. As a result, the discretization erroneously mixed the
cool water at the surface with the underneath warm water using a much larger diffusion
coefficient. Hence, using this discretization, the model failed to reproduce the cool-skin effect.
In contrast, the cool-skin effect was well reproduced using the discretizations W01T10S02
and W05T10S02, where the uppermost numerical heat fluxes were determined at 0.25 mm.
Note that the depth of 0.25 mm is at the middle point of the conduction layer. By comparing
the case of W01T10S02 with that of W01T50S02, it can be seen that failure to reproduce the
cool-skin effect causes overestimates of latent heat flux, upward sensible heat flux, and
terrestrial radiation by 5.5, 1.3 and 0.8 W m-2, respectively, with a sum of 7.6 W m-2.
5.2
Warm-layer Discretization
As can be seen in Figure 4c, the warm-layer effect was underestimated using the
discretization W05T10S02, where the warmest temperature was determined at a depth of 2.5
m at 3 PM. The depth of 2.5 m is lower than the warmest layers between the depths of 1mm
and 1m as described earlier (Figure 4a). In contrast, the warm-layer phenomena were well
reproduced using the discretizations W01T10S02 and W01T50S02, where the warmest
temperature was determined at 0.5 m, between the depths of 1 mm and 1 m, at 3 PM. By
comparing the fluxes between W01T10S02 and W05T10S02, it can be seen that failure to
reproduce the warm layer causes underestimates of latent heat flux, upward sensible heat flux,
and terrestrial radiation to the atmosphere by 0.5, 0.13 and 0.03 W m-2, respectively, with a
20
sum of 0.7 W m-2.
5.3
Time Lag
As described in Eq. (5) and Eq. (6), the suggested skin-layer parameterization
theoretically suffers a phase error between calculated and observed SST. Figure 7 compares
the composite hourly means of SST using various vertical resolutions. There is a 1-h time lag
for the highest SST simulated using the finest discretization W01T01S27, and a 3-h time lag
using the other coarser discretizations. This suggests that an even finer discretization might
ameliorate the time lag problem, but it will make the scheme even more expensive.
6 Conclusion
Overall, using the TOGA data as a case study, the TKE approach plus skin-layer
parameterization (SLP) developed here have demonstrated the capability of the model to
reproduce SST and temperature profiles of water near the ocean surface. This shows that the
cool skin and the warm layer can only be simulated by a proper discretization of a water
column within the top 1-m depth. To capture the cool-skin effect, at least two temperatures
simulated within the conduction layer are required. One at the surface and the other at the
bottom of the conduction layer are suggested. The thickness of the conduction layer at the
study site is found to be 0.5 mm. In order to capture the warm-layer effect, temperatures
simulated at depths between 1 mm – 1 m are required. We have verified this concept by
applying it to the data taken at the study site, using a fine vertical discretization W01T01S27,
21
which has 27 numerical layers within the top 1-m depth. The bias of ∆T3m between
simulations and observations is 0.03 K, with a correlation of 0.57 and a RMSE of 0.27 K.
Nonetheless, having 27 numerical layers within the top 1-m depth is computationally
expensive. Alternatively, the discretizations W01T10S02 and W02T10S02 show economical
ways for simulating both the cool-skin and the warm-layer phenomena with accuracy similar
to W01T01S27. In W01T10S02 and W02T10S02, only 2 layers are required within the top
1-m depth. Both of them determine the fluxes at depths of 0.25 mm and 1 mm. W01T10S02
determines the temperatures at the surface, at 0.5 mm and at ~ 0.5 m, and W02T10S02
determines the temperatures at the surface, at 0.5 mm and at ~ 1 m.
The SLP presented here can be organized as a tri-diagonal matrix, which can be solved
implicitly without the constraints of numerical stability. Nonetheless, the SLP suggested here
still suffers from a time lag of 1 - 3 h for peak skin temperature simulation, depending on the
resolution chosen.
Acknowledgement
The author would like to acknowledge Dr. J.-M. Lefevre for kindly offering the code of
Gaspar et al. (1990). Suggestions from Prof. Wu, C.-C. at Nat’l Taiwan Univ. and Dr. E.
Bauerle at U. of Constance on a finer vertical resolution in the upper surface are helpful.
Encouragement of the work by Dr. L. Dumenil-Gate, Dr. K. Arpe and Dr. L. Bengtsson are
appreciated. Also thanks give to M. Tucker and Dr. N. Keenlyside for proofreading. This
22
work is supported by National Science Council/Taiwan and Deutscher Akademischer
Austauschdienst Programmabteilung/Germany exchange program under contracts 35110F,
86-2621-P-005-019, A/98/19488, 91-2111-M-005-001 and 92-2111-M-005-001.
Appendix - Numerical Method
A numerical grid is chosen to discretize the water column as shown in Figure A1. That is,
T, U, and S are determined at the center of the grid, and fluxes T ' w ' , U ' w ' , S ' w ' and
turbulent kinetic energy E are determined at the boundaries of the grid. Eqs. (17)-(19) and (28)
for all the layers except that k = 0, 1 and g. The skin layer and the first layer are
parameterized according to Eq. (12) and Eq. (16), respectively. The resulting temperature
matrix is:
⎡1 + βy 0
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
− βy 0
h
− βx1 − e
h1
1 + βx1 + βy1
− βy1
− βx k
1 + βx k + βy k
− βy k
− βx g
⎤ ⎡T j +1 ⎤
⎥ ⎢ 0 j +1 ⎥
⎥ ⎢T1 ⎥
⎥⎢
⎥
⎥⎢
j +1 ⎥
⎥ ⎢Tk −1 ⎥
⎥ ⎢ j +1 ⎥
⎥ ⎢Tk ⎥ =
⎥ ⎢Tk j++11 ⎥
⎥⎢
⎥
⎥⎢
⎥
⎥ ⎢T j +1 ⎥
⎥ ⎢ nj +1 ⎥
1 + βx g ⎥⎦ ⎣Tg ⎦
∆t ⋅ G0 ∆t ⋅ Rsn [F ( z 0 ) − F ( z 0 − d )]
⎡
− (1 − β ) y 0 T0 j − T1 j
T0 j +
+
⎢
ρ w c w he
ρ w c w he
⎢
⎛ z1 + z 2 ⎞ ⎤
⎢ j he j ∆t ⋅ Rsn ⎡
j
j
j
j
⎢T1 − h T0 + ρ c h ⎢ F ( z 0 − d ) − F ⎜⎝ 2 ⎟⎠⎥ + (1 − β ) x1 T0 − T1 − y1 T1 − T2
⎦
1
w w 1 ⎣
⎢
⎢
⎢
⎢
⎢T j + ∆t ⋅ Rsn ⎡ F ⎛⎜ z k −1 + z k ⎞⎟ − F ⎛⎜ z k + z k +1 ⎞⎟⎤ + (1 − β ) x T j − T j − y T j − T j
k
k −1
k
k
k
k +1
⎥
⎢ k
ρ w c w hk ⎢⎣ ⎝
2
2
⎠
⎝
⎠⎦
⎢
⎢
⎢
⎢
⎢
∆tRsn F (z g )
⎢
+ (1 − β )x g Tnj − Tgj
Tgj +
⎢
kg
⎢
ρ g cg
ω
⎣⎢
(
)
[ (
)
[ (
(
)
)
23
(
(
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦⎥
)]
)]
(A1)
where ρ g , c g are the soil density (kg m-3) and the specific heat (J kg-1 K) at the bed of the
water column, respectively. For shallow water all the solar radiation reaching the waterbed is
assumed to be absorbed by the waterbed without any reflection. Variables in Eq. (A1) are
described as follows:
y0 =
∆t k0
he z0 − z1
xk =
∆t ⎛ k k −1
⎜
hk ⎜⎝ z k −1 − z k
⎞
⎟⎟ for k = 1, n
⎠
(A2b)
yk =
⎞
∆t ⎛ k k
⎜⎜
⎟ for k = 1, n
hk ⎝ z k − z k +1 ⎟⎠
(A2c)
xg =
∆tρ w c w ⎛⎜ k n
k g ⎜⎝ z n − z g
ρ g cg
(A2d)
(A2a)
⎞
⎟
⎟
⎠
ω
and
he =
⎛ 0.25h1
k 0 ⎛⎜
1 − exp⎜ −
⎜
ω ⎜⎝
2k 0 / ω
⎝
⎞⎞
⎟⎟
⎟⎟
⎠⎠
(A3a)
h1 = z 0 − 0.5( z1 + z 2 )
(A3b)
hk = 0.5( z k −1 − z k +1 ) for k = 2, n
(A3c)
Note that β=0 for the forward scheme, β=0.5 for the Crank-Nicolson scheme, and β=1 for the
backward scheme. The backward scheme is found to be desirable since it is numerical
unconditionally stable, and the backward scheme is used in the case study. The matrix is a
tri-diagonal matrix and can be easily and efficiently solved by the LU method (e.g. Press et
al., 1992). Similar matrixes can be derived for U and S.
24
The TKE equation (28) is solved by linearizing it as:
⎛
1/ 2 ⎞
c
⎜⎜1 + 1.5 ε Ekj ⎟⎟ Ekj +1 + β xk Ekj +1 − Ekj +1 − y k E kj +1 − E kj++11 =
lε
⎝
⎠
j
Ek − (1 − β ) xk Ekj−1 − Ekj − y k E kj − Ekj+1
[ (
[ (
)
)
(
)]
)]
(
2
⎡
⎛ U kj − U kj+1 ⎞
g ⎛ ρ kj − ρ kj+1 ⎞⎤
⎜
⎟⎥
⎟⎟
+ k k ∆t
+ β 2 ⎢k k ∆t ⎜⎜
ρ w0 ⎜⎝ z k − z k +1 ⎟⎠⎥
⎢⎣
⎝ z k − z k +1 ⎠ i =u ,v
⎦
(A4)
2
⎡
⎛ U kj +1 − U kj++11 ⎞
g ⎛ ρ kj +1 − ρ kj++11 ⎞⎤
⎟⎥
⎜
⎟⎟
+ k k ∆t
+ (1 − β 2 )⎢k k ∆t ⎜⎜
ρ w0 ⎜⎝ z k − z k +1 ⎟⎠⎥
⎢⎣
⎝ z k − z k +1 ⎠ i =u ,v
⎦
3/ 2
Ej
− cε k
2lε
where β 2 is set to be 0.5.
Reference
Adamec, D., R.L. Elsberry, R.W. Garwood, and R.L. Haney, 1981: An
embedded mixed-layer-ocean circulation model, Dyn. Atmos. Ocean , 6,
69-77.
Arakawa, A. and Y. Mintz, 1974: The UCLA atmospheric general circulation
model. Notes distributed at the workshop 25 March – 4 April 1974,
Department of Meteorology, University of California, Los Angeles,
California 90024.
Bougeault, P. and Lacarrere, 1989: Parameterization of orography-induced
turbulence in a meso-beta scale model, Mon. Weather Rev. , 117 ,
1872-1890.
Briscoe, M.G., and R.A. Weller, 1984: Preliminary results from the Long
Term Upper Ocean Study (LOTUS), Dyn. Atmos. Ocean , 8, 243-265.
Brutsaert, W., 1982, Evaporation into the Atmosphere. Reidel Pub. Co., pp.
299.
Businger, J.A., J.C. Wyngaard, Y. Izumi and E.F. Bradley, 1971: Flux-Profile
Relationships in the Atmospheric Surface Layer, J. the Atmospheric
Sciences., 28 , 181-189.
Carslaw, H.S. and J.C. Jaeger, 1959: Conduction of Heat in Solids . 2 nd.,
Oxford Press, 509 pp.
Castro, S. L.; Wick, G. A.and Emery, W. J., 2003: Further refinements to
models for the bulk ­ skin sea surface temperature difference. J. Geophys.
Res., Vol. 108, No. C12, 3377.
Chia, H.H. and Wu, C.-C., 1998: Air-sea eddy fluxes and the mixed layer of
25
the western equatorial pacific: observation and one-diemnsional model
simulation. Atmospheric Sciences , 26 (2), 157-179 (in Chinese).
Dalu, G.A. and Purini, R., 1982: The diurnal thermocline due to buoyant
convection. Quart. J. R . Met. Soc. , 108 , 929-935.
Davis, R.E., R. de Szoeke, D. Halpern, and P. Niiler, 1981: Variability in the
upper ocean during MILE, I, The heat and momentum balance, Deep Sea
Res., Part A, 28, 1427-1451.
Deardorff, J.W., 1978: Efficient prediction of ground surface temperature and
moisture, with inclusion of a layer of vegetation. J. Geophys. Res., 83 ,
1889-1903.
Donlon, C.J., P.J. Minnet, C. Gentemann, T.J. Nightingale, I.J. Barton, B.
Ward and M.J. Murray, 2002: Toward improved validation of satellite sea
surface skin temperature measurements for climate research. J. Climate ,
15 , 353-369.
Fairall, C.W., Bradley, E.F., Godfrey, J.S., Wick, G.A., Edson, J.B., and
Young, G.S., 1996: Cool-skin and warm-layer effects on sea surface
temperature. J. Geophys. Res., 83 , 1889-1903.
Garratt, J.R., 1977: Review of drag coefficients over oceans and continents.
Mon. Weather Rev. , 105 , 915-929.
Garwood, R.W., 1977: An oceanic mixed-layer model capable of simulating
cyclic states, J. Phys. Oceanogr., 7, 455-471.
Gaspar, P., Y. Gregoris and J.-M. Lefevre, 1990: A simple eddy kinetic
energy model for simulations of the oceanic vertical mixing: Test at
station Papa and long-term upper ocean study site. J. Geophys. Res. , 95 ,
16179-16193.
Gates, W.L., 1992: The atmospheric model intercomparison project. Bull. Am.
Meteorol. Soc. , 73 , 35-62.
Grassl, H., 1976: The dependence of the measured cool skin of the ocean on
wind stress and total heat flux. Bounday-Layer Meteorol. , 10 , 465-474.
Gregg, M.C., Peters, H., Wesson, J.C., Oakey, N.S. and Shay, T.J., 1985:
Intensive measurements of turbulence and shear in the equatorial
undercurrent. Nature , 318 , 140-144.
Haidvogel, D.B., and F.O. Bryan, 1992: Ocean general circulation modeling.
pp. 371- 412, in Climate System Modeling , edited by K.E. Trenberth,
Cambridge university press, 788 pp.
Hasse, L., 1971: The sea surface temperature deviation andd the heat flow at
the sea-air interface. Bounday-Layer Meteorol. , 1 , 368-379.
Horrocks, Lisa A.; B. Candy, T. J. Nightingale,R. W. Saunders, A. O'Carroll
and A. R. Harris, 2003: Parameterizations of the ocean skin effect and
implications for satellite-based measurement of sea-surface temperature.
J. Geophys. Res. Vol. 108 No. C3 3096.
26
Hostetler, S.W., G.T. Bates, and F. Giorgi, 1993: Interactive coupling of a
lake thermal model with a regional climate model. J. Geophys. Res. , 98 ,
5045-5057.
Hong, X., S. Raman, R.M. Hodur and L. Xu, 1999: The Mutual Response of
the Tropical Squal Line and the Ocean. Pure Appl. Geophys. , 155 , 1-32.
Ineson, S., and C. Gordon, 1989: Parameterization of the upper ocean mixed
layer in a tropical ocean GCM, Dyn. Climatol. Tech. Note , 74 , Meteorol.
Office, Bracknell, Berkshire, England.
Jerlov, N.G., Marine Optics , Elsevier Oceanogr. Ser. , 14, 231 pp., Elsevier,
New York, 1976.
Khundzhua, G.G., Gusev, A.M., Andreyev, E.G., Gurov, V.V. and
Skorokhvotov, N.A., 1977: About the structure of surface cold film of
the ocean. Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana. , 13 , 753-758.
Leredde, Y., Devenon, J.-L., and Dekeyser, I., 1999: Turbulent viscosity
optimized by data assimilation. Ann. Geophysicae , 17 , 1463-1477.
Kley, D., H. G. J. Smit, H. Vomel, H. Grassl, V. Ramanathan, P J. Crutzen, S.
Williams, J. Meywerk, and S. J. Oltmans, 1997: Tropospheric
water-vapour and ozone cross-sections in a zonal plane over the central
equatorial Pacific Ocean, Quarterly Journal of the Royal Meteorological
Society , 123: (543) 2009-2040 Part A OCT
Levitus, S., Climatological atlas of the world ocean, Prof. Pap. 13 , 173 pp,
Natl. Oceanic and Atmos. Admin., Boulder, Colo., 1982.
Martin, P.J., 1985: Simulation of the mixed layer at OWS November and Papa
with several models. J. Geophys. Res. , 90 , 903-916.
Mellor, G.L. and P.A. Durbin, 1975: The strucrure and dynamics of the ocean
surface mixing layer. J. Phys. Oceanogr. , 5 , 718-728.
Mellor, G.L., and T. Yamada, 1982: Development of a turbulence closure
model for geophysical fluid problems. Rev. Geophys. Space Phys. , 20 ,
851-875.
Niiler, P.P., 1975: Deepening of the wind-mixed layer, J. Mar. Res., 33 ,
405-422.
Oberhuber, J. M., 1993: Simulation of the Atlantic circulation with a coupled
sea ice-mixed layer-isopycnal general circulation model. Part I: Model
description. J. Phys. Oceanogr. , 23 (5), 808-829.
Oberhuber, J. M., 1993: Simulation of the Atlantic circulation with a coupled
sea ice-mixed layer-isopycnal general circulation model. Part II: Model
experiment. J. Phys. Oceanogr. , 23 (5), 830-845.
Paulson, C.A., J. J. Simpson, 1981: The Temperature Difference Across Cool
Skin of the Ocean. J. Geophys. Res., 86, C11, 11,044-11,054.
Peters, H., Gregg, M.C., and Toole, J.M., 1988: On the parameterization of
27
equatorial turbulence, J. Geophys. Res., 93 (C2) , 1199-1218.
Pielke, R.A., 1984: Mesoscale Meteorological Modeling, Academic Press,
Orlando, Florida, 612pp.
Pinazo C, Marsaleix P, Millet B, Estournel, C., and Vehil, R., 1996. Spatial
and temporal variability of phytoplankton biomass in upwelling areas of
the northwestern Mediterranean: A coupled physical and biogeochemical
modelling approach. J. Marine Syst. , 7, 161-191
Press, W.H., S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, 1992:
Numerical Recipes in Fortran . Cambridge University Press, 963 pp.
Price, J.F., R.A. Weller, C.M. Bowers, and M.G. Briscoe, Diurnal response of
sea surface temperature observed at the Long-Term Upper Ocean Study
(34°N, 70°W) in the Sargasso Sea, J. Geophys. Res., 92, 14,480-14,490,
1987.
Saunders, P.M., 1967: The temperature at the ocean-air interface, J. Atmos.
Sci. , 24 , 269-273.
Schanz, L. C., and P. Schlussel, 1997: Atmospheric back radiation in the
tropical Pacific: intercomparison of in-situ measurements, simulations
and satellite retrievals, Meteorology and Atmospheric Physics , 63(3-4):
217-226
Schopf, P.S., and M.A. Cane, 1983: On equatorial dynamics, mixed layer
physics and sea surface temperature, J. Phys. Oceanogr. , 13 , 917-935.
Soloviev, A.V. and Schlussel, P., 1994: Parameterization of the cool skin of
the ocean and of the air-ocean gas transfer on the basis of modelling
surface renewal, J. Phys. Oceanogr. , 24 , 1339-1346.
Soloviev, A.V. and Schlussel, P., 1996: Evolution of cool skin and direct
air-sea gas transfer coefficient during daytime. Bounday-Layer Meteorol. ,
77 , 45-68.
Stramma, L., P. Cornillon, R.A. Weller, J. Price, and M.G. Briscoe, 1986:
Large diurnal sea surface temperature variability: Satellite and in situ
measurements, J. Phys. Oceanogr., 13, 1894-1907.
Sun, N.-Z, 1994: Inverse Problem in Groundwater Modeling, Kluwer Acad.,
Norwell, Mass. , 337.pp.
Tsuang, B.-J., and J.A. Dracup, 1991: Evaporation rates from
temperature-startified saline lakes using a one-dimensional integrated
evaporation methodology - Mono Lake as a case study, California Water
Resources Center, University of California Contribution 202, 76pp.
Tsuang, B.-J., and H.-C. Yuan, 1994: The ideal numerical surface thickness
to determine ground surface temperature and schemes comparison.
Atmospheric Sciences , 21 , 189-218 (in Chinese).
Tsuang, B.-J.; Tu, C.-Y., 2002: Model structure and land parameter
identification: an inverse problem approach. J. Geophys. Res. –
28
Atmospheres, 107 (D10), 4096, doi:10.1029/2001JD000711.
Tsuang, B.-J., 2003: Analytical asymptotic solutions to determine
interactions between the planetary boundary layer and the Earth's surface.
J. Geophys. Res. – Atmospheres, 108 (D16), 8608,
doi:10.1029/2002JD002557.
UNESCO, 1981: UNESCO Tech. Pap. Mar. 36 , 13.
Webster, P.J., and R. Lukas, 1992: TOGA COARE: The coupled
ocean-atmosphere response experiment, Bull. Am. Meteorol. Soc. , 73 ,
1377-1416.
Wells, N.C., 1979: A coupled ocean-atmosphere experiment: The ocean
response, Q. J. R. Meteorol. Soc., 105, 355-370.
Wick, G. A., W. J. Emery, L. H. Kantha, and P. Schlussel, 1996: The
behavior of the bulk-skin temperature difference under varying wind
speed and heat flux. J. Phys. Oceanogr. , 26, 1969–1988.
Wu, J., 1985: On the cool skin of the ocean. Bounday-Layer Meteorol. , 31 ,
203-207.
29
z
t = t0
t = t0 + ∂t
∂T0
x
T0
G0
he =
d=h1/4
h1
G01
G12
T1
T (z , t ) = T0 + ∆T0 exp
k⎛
d
⎛
⎜1 − exp ⎜ −
ω ⎜⎝
2k / ω
⎝
⎞⎞
⎟ ⎟⎟
⎠⎠
≈ he∂T0
(
(ω / 2k )z )cos(ω (t − tm ) + z /
2k / ω z
)
T2
Figure 1 Schematic of effective skin layer thickness he of an ideal water surface with depth d.
T0 is the skin temperature and T1 is the average temperature of the uppermost layer of
the water with a thickness of h1. Note that the shaded area denotes the energy stored
from the surface to depth d, which is close to he ∂T0 .
30
Rsn ( W/m 2 )
(a)
1200
600
0
(b)
G0 ( W/m 2 )
1200
600
0
(c)
WS (m/s)
20
15
10
5
0
(d)
SST ( oC )
31
30
obs
cal
obs
cal
obs
cal
29
28
(e)
T3m ( oC )
31
30
29
28
∆T3m ( K )
(f)
1
0
-1
1
3
5
7
9
11
February 1993
31
13
15
17
19
21
Figure 2 Time series of (a) solar radiation (Rsn), (b) net upward non-solar flux (G0), (c)
wind speed (WS), (d) skin sea surface temperature (SST), (e) temperature at a 3-m depth
(T3m) and (f) ∆T3m (SST-T3m).
32
(b) daytime
(c) nighttime
1.5
1.5
1.0
1.0
1.0
0.5
0.0
-0.5
∆T3m (K)
1.5
∆T3m (K)
∆T3m (K)
(a) daytime
0.5
0.0
-0.5
-1.0
400
800
Rsn ( W/m2 )
1200
0.5
0.0
-0.5
-1.0
0
obs
cal
-1.0
0
10
WS ( m/s )
20
0
10
WS ( m/s )
20
Figure 3 Observed (solid rhombus) and simulated (open circle) ∆T3m as a function of (a)
solar irradiance during daytime, (b) wind speed during nighttime, and (c) wind speed
during daytime.
33
(a) W01T01S27
0.00001
0
Depth (m)
0.0001
0.001
0.01
6
3
0
9
21
18
12
15
0.1
1
10
(b) W01T10S02
0.00001
0
Depth (m)
0.0001
0.001
6
3 9
0
21
12
15
18
0.01
0.1
1
10
(c) W05T10S02
0.00001
0
Depth (m)
0.0001
0.001
0AM
3
6
conduction
layer
9
12
6 9 3 0
21 12 18 15
0.01
0.1
1
10
(d) W01T50S02
34
15
viscous layer
18
21
0AM
3
6
9
12
15
18
21
0.00001
0
Depth (m)
0.0001
0.001
6
0.01
3 9
0
21
18
12
15
0.1
1
10
28.8
29.0
29.2
29.4
o
Temperature ( C)
29.6
29.8
Figure 4 Composite diurnal profiles of near surface temperature simulated using the
discretizations of (a) W01T01S27, (b) W01T10S02, (c) W05T10S02 and (d)
W01T50S02, where lines denote simulations, symbols denote observations.
35
Bias and RMSE (K) of ∆T 3m
Bias
0.5
RMSE
0.0
-0.5
1.0E-05
1.0E-04
1.0E-03
Thickness of Conduction Layer (m)
1.0E-02
Figure 5 Bias (calculation–observation) and root-mean-square-error (RMSE) of ∆T3m during
nighttime as functions of the thickness of the conduction layer.
36
(a)
∆T3m ( K )
1
0
Obs
-1
W01
W02
W05
(b)
Obs
∆T3m ( K )
1
W01T10S02
W02T10S02
W05T10S02
0
-1
1
3
5
7
9
11
13
15
17
19
21
February 1993
Figure 6 Comparisons of simulated ∆T3m between (a) conventional discretizations (W01,
W02 and W05) and (b) suggested discretizations (W01T10S02, W02T10S02 and
W05T10S02), which include the skin layer and the conduction layer.
37
Obs
W01T01S27
W01T10S02
W02T10S02
W05T10S02
W10T10S02
29.4
o
SST ( C)
29.7
29.1
28.8
0
6
12
18
Local Time
Figure 7 Composite means of hourly SST of observations and simulations using various
discretizations.
38
z
T0,S0,U0
h1
0
x
T1,S1,U1
1
zk
T2,S2,U2
2
…..
k-1
hk
Tk,Sk,Uk
zg=-D
k
E k , T ' w'k , S ' w'k , U ' w'k
…..
Tn,Sn,Un
Tg,Sg,Ug
n-1
n
Figure A1
Grid structure of the model, where D is the depth of a water column (m). Note
that layers from 2 to n are conventional numerical grids. In addition, a skin layer for
water surface k=0 and a soil layer k=g are added. T1 represents the average temperature
of the first layer (from surface to depth h1).
39
Table 1. Vertical discretizations discussed in this study
Name*
skin
0<x<1m
1 m ≤ x ≤ 10 m
layer
100 µm, 200 µm, 300 µm, 400 µm, 500 µm, 600 µm,
W01T01S27
0 µm
W01
no
W01T10S02
0 µm
1 mm
W01T50S02
0 µm
5 mm
W02
no
W02T10S02
0 µm
W05
no
W05T10S02
0 µm
700 µm, 800 µm, 900 µm, 1 mm, 2 mm, 3 mm, 5 mm, 1
cm 2 cm 3 cm 5 cm 10 cm 20 cm 30 cm 40 cm 50
-
1 m, 2 m, 3 m, 4 m, 5 m, 6 m,
7 m, 8 m, 9 m, 10 m
2 m, 4 m, 6 m, 8 m, 10 m
1 mm
5 m, 10 m
1 mm
40
Table 2. Statistics of simulated SST (oC) and △T3m (K) by using different discretizations
ation
Mean Corr.
△T3m (night)
△T3m (all)
SST (all)
Discretiz
RMSE Mean. Corr.
RMSE Mean. Corr.
△T3m (day, C.S.)
RMSE Mean. Corr.
△T3m (day, W.L.)
RMSE Mean. Corr.
RMSE
29.19
-
-
-0.13
-
-
-0.12
-
-
-0.21
-
-
0.21
-
-
29.15
0.79
0.26
-0.10
0.57
0.27
-0.10
0.51
0.27
-0.19
0.21
0.22
0.34
0.21
0.48
W01
29.39
0.72
0.37
0.19
0.39
0.47
0.06
0.08
0.40
0.26
0.28
0.52
0.57
0.28
0.58
W01T10S02
29.17
0.70
0.26
-0.08
0.42
0.26
-0.08
0.41
0.26
-0.14
0.15
0.23
0.17
0.15
0.36
W01T50S02
29.29
0.73
0.26
0.05
0.43
0.30
-0.02
0.05
0.27
0.08
0.33
0.31
0.37
0.33
0.42
W02
29.36
0.65
0.31
0.12
0.35
0.34
0.05
0.08
0.33
0.16
0.25
0.35
0.31
0.25
0.31
W02T10S02
29.17
0.64
0.26
-0.12
0.40
0.21
-0.12
0.38
0.21
-0.15
0.17
0.18
0.06
0.17
0.31
W05
29.31
0.55
0.27
0.01
0.31
0.24
0.01
0.19
0.25
0.01
0.21
0.21
0.03
0.21
0.27
W05T10S02
29.11
0.58
0.26
-0.13
0.28
0.18
-0.13
0.26
0.18
-0.14
0.17
0.13
-0.10
0.18
0.36
Observation
W01T01
Here: “all” for entire period, “night” for zero solar irradiance period, “day” for non-zero solar irradiance period, “C.S.” for cool-skin
prevailing (△T3m<0) period, and “W.L.” for warm-layer prevailing (△T3m>0) period.
41