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Appendix
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Resolving the upper-ocean warm layer improves the simulation of the
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Madden-Julian Oscillation
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Wan-Ling Tseng1,2, Ben-Jei Tsuang3, Noel S. Keenlyside4, Huang-Hsiung Hsu1, &
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Chia-Ying Tu1
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1
Research Center for Environmental Changes, Academia Sinica, Taipei, Taiwan.
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2
GEOMAR | Helmholtz-Zentrum für Ozeanforschung, Kiel, Germany.
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3
National Chung-Hsing University, Taichung, Taiwan.
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4
Geophysical Institute and Bjerknes Centre, University of Bergen, Bergen, Norway.
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Corresponding author: W.-L. Tseng, Research Center for Environmental Changes,
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Academia Sinica, Taipei, 115, Taiwan. (wtseng@gate.sinica.edu.tw)
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TEL：886-2-2652-5174
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FAX：886-2-2783-3584
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The one-column ocean model, Snow-Ice-Thermocline (SIT), is based on Tsuang
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et al. (2001) that follows the turbulent kinetic energy (TKE) approach of Gaspar et al.
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(1990). SIT parameterizes ice-formation, the warm-layer effect and the cool-skin
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effect, and introduces a surface effective thickness (ℎ𝑒 ) to improve the simulation of
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upper ocean temperature (Tu and Tsuang 2005; Tsuang et al. 2009). The model has
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been verified at a tropical ocean site (Tu and Tsuang 2005), in the South China Sea
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(Lan et al. 2010), and in the Caspian Sea (Tsuang et al. 2001). The melt and formation
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of snow and ice above a water column has been introduced (Tsuang et al. 2001).
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However, these parts are not utilized in the experiments here.
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SIT determines water temperature, salinity, and u, v currents (denoted as
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variable X) in each depth according to the energy, salinity and momentum budget in a
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one-column model (e.g., Gaspar et al., 1990) as:
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𝜕𝑋̅
𝜕𝑧
=−
̅̅̅̅̅̅̅
𝜕𝑋′𝑤′
𝜕𝑧
(1)
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where X ' w' (positive upward) is the vertical flux of scalar X, a transport property. It
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can be temperature (T) (K), horizontal velocities (u, v) (m s-1) or salinity (S) (practical
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salinity ‰). Variable z is the height (positive upward). The over bar represents a time
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averaged value. Furthermore, the penetration of solar radiation is parameterized
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according to a nine-band equation (Paulson and Simpson 1981), where the absorption
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coefficients of solar radiation are set according to (Fairall et al. 1996).
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To simulate the warm layer the vertical resolution of the water column needs to
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be fine enough to resolve variations in the upper 10 m of the ocean. Below the cool
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skin, the vertical flux X ' w' is parameterized using the classical K approach as:
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𝜕𝑋̅
̅̅̅̅̅̅
𝑋′𝑤′ = −(𝑘 + 𝜈) 𝜕𝑧
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where k and ν are eddy and molecular diffusion coefficients (m2 s-1), respectively.
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Furthermore k and ν are designated as km and νm for momentum, and as kh and νh for
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temperature and salinity. The surface net heat flux (latent heat flux + sensible heat
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flux + net long wave radiation) are used as the upper boundary condition for heat Eq;
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the wind stress is used as the upper boundary condition for momentum Eq; the net
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fresh water salinity flux, (evaporation - precipitation - river inflow) multiply by
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surface salinity, is used as the upper boundary condition for salinity Eq.
(2)
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To simulate the cool skin effect, the eddy diffusion coefficient for heat, kh,
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within the cool skin and the eddy diffusion coefficient for momentum, km, within the
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viscous layer are set to zero: molecular transport is the only mechanism for vertical
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diffusion of heat and momentum in the cool skin and in the viscous layer, respectively
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(Hasse 1971; Grassl 1976; Wu 1985). The molecular diffusion coefficient for
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momentum, 𝜈𝑚 , is set at 1.20×10-6 m2 s-1, and that for heat, 𝜈ℎ , is set at 1.34×10-7 m2
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s-1, according to (Paulson and Simpson 1981). The thickness of the cool skin δ is
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determined by (Saunders 1967) as:
𝜆𝜈𝑚
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𝛿=
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where λ is a dimensionless constant and 𝑢∗ is the friction velocity of water (m/s). SIT
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determines λ according to (Artale et al. 2002). Below the cool skin and the viscous
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layer, eddy diffusivity is determined according to a TKE-mixing length approach
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(Gaspar et al. 1990).
𝑢∗
(3)
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To correct bulk SST computed using conventional discretization to skin SST we
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introduce a surface effective thickness (ℎ𝑒 ) (Tu 2006; Tu and Tsuang 2014). The
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effective thickness is a function of the surface layer, which is added to the top of the
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uppermost numerical layer of the conventional discretization (Fig. A1). This surface
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layer has a physical thickness d of 0.25 h1, i.e., d = 0.25 h1. The net heat flux absorbed
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within the surface layer is G0+Rsn[F(z0)-F(z0-d)]+G0,1. To determine the upper skin
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temperature T0 (not the column-mean temperature) of the surface layer, T0 is
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parameterized as:
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 w cw he
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Where he is the effective thickness (m) for heat of the surface layer. Note that the first
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term on the right-hand side of the above equation is a cooling term since the non-solar
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surface heat flux G0 is usually upward (Saunders 1967; Dalu and Purini 1982;
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Soloviev and Schlüssel 1994; Fairall et al. 1996). The second term is a heating term
T0
 G0  Rsn F z0   F z0  d   G0,1
t
(4)
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due to the absorption of shortwave solar radiation (Fairall et al. 1996). The third term
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is the vertical heat flux due to molecular (if within the skin layer) or eddy (if below
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the skin layer) diffusivity. Eq. (4) is proposed by this study to calculate the skin
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temperature T0. The effective thickness is a function of heat conductivity. It is derived
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analytically to reproduce the diurnal fluctuation of skin temperature if the fluctuation
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can be described as a cosine function in time (Tsuang et al. 2009). The effective
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thickness is derived to be:
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

 
  
 


 
  
 


k 
d   d   
d   d 

he 
1  exp 
cos
 exp 
sin

 
2k   2k    
2k   2k  

 
  
 


      
    


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where  is the angular velocity of the earth with respect to the sun (=2/86400 s-1).
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Note that the upper temperature T0 of the skin layer is the so-called sea surface
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temperature (SST). Once SST is determined, we can determine heat fluxes between
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the atmosphere and ocean for the next model timestep. Then, the error due to incorrect
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usage of T1 for T0 to determine heat exchange between the atmosphere and ocean in
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the conventional approach is corrected.
2
2
(5)
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Overall, SIT simulates the SST and upper ocean temperature variations, including
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the cool-skin and warm-layer mechanisms. In the finest resolution experiments, SIT
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has 42 vertical layers, and with 12 in the upper 10m. Simulated water temperatures
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are at surface, and grid cells with center at depth of 0.05mm, 1 m, 2 m, 3 m, 4 m, 5 m,
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6 m, 7 m, 8 m, 9 m, 10 m, 16.8 m, 29.5 m, 43.6 m, 59.3 m, 76.9 m, 96.8 m, 119.4 m,
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145.3 m, 174.9 m, 208.9 m, 248.3 m, 293.8 m, 346.8 m, 408.4 m, 480.2 m, 564.3 m,
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662.6 m, 779.7 m, 913.1 m, 1072 m, 1258.8 m, 1478.6 m, 1737.3m, 2042m, 2401m,
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2824.4m, 3323.6m, 3912.4m, 4607.1m and the ocean seabed. The resolution in the
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upper 10 m is very fine in order to capture the upper ocean warm layer, and there is a
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layer at 0.05 mm and resolving a corresponding effective thickness for reproducing
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the cool skin of the ocean surface. For the C-17m we deleted layer from 0.05mm to 10
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m and C-59m we deleted layer from 0.05mm to 43.6 m.
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In addition, a nudging technique is used to correct the bias of calculated ocean
temperature and salinity (denoted as X) at layers deeper than 10 m depth as:
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∗,𝑛+1
∗,𝑛+1
𝑛+1
𝑛+1
𝑋𝑘,𝑐
= 𝑋𝑘,𝑐
+ 𝛽(𝑋𝑘,𝑜
− 𝑋𝑘,𝑐
)
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∗,𝑛+1
𝑛+1
where 𝑋𝑘,𝑐
is calculated X at depth k at timestep n+1; 𝑋𝑘,𝑐
is calculated 𝑋𝑘,𝑐 by
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𝑛+1
Eq. (1) at timestep n+1; 𝑋𝑘,𝑜
is observed 𝑋𝑘,𝑐 at timestep n+1; 𝛽 is a relaxation
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factor. It should be within 0 and 1. When setting 𝛽 at 0, the calculated X is determined
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by Eq. 1 only; when setting 𝛽 at 1, the calculated X is restored back to observed X
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every time step. The relaxation factor is parameterized as:
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𝛽 = 0.5 𝜏
(6)
∆𝑡
(7)
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Where 𝜏 is the timescale for nudging; ∆𝑡 is the time step. To account for neglected
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horizontal processes, the ocean is weakly nudged with a 30-day time scale for depths
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within 10-100 m (i.e., τ = 30 𝑑), and 1-day time scale for depths > 100 m (i.e., τ =
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1 𝑑) to the observed climatological ocean temperature; there is no nudging within the
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upper 10-m depth.
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Reference
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Artale V, Iudicone D, Santoleri R, Rupolo V, Marullo S, D'Ortenzio F (2002) Role of
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surface fluxes in ocean general circulation models using satellite sea surface
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temperature: Validation of and sensitivity to the forcing frequency of the
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Mediterranean thermohaline circulation. J Geophys Res 107 (C8):29-21-29-24
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Dalu G, Purini R (1982) The diurnal thermocline due to buoyant convection.
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Quarterly Journal of the Royal Meteorological Society 108 (458):929-935
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Fairall C, Bradley EF, Godfrey J, Wick G, Edson JB, Young G (1996) Cool-skin and
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warm-layer effects on sea surface temperature. Journal of Geophysical
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research 101 (C1):1295-1308
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Gaspar P, Gregoris Y, Lefevre J-M (1990) A simple eddy kinetic energy model for
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simulations of the oceanic vertical mixing: Tests at station Papa and long-term
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upper ocean study site. J Geophys Res 95 (C9):16179-16193
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Grassl H (1976) The dependence of the measured cool skin of the ocean on wind
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stress and total heat flux. Boundary-Layer Meteorology 10 (4):465-474
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Hasse L (1971) The sea surface temperature deviation and the heat flow at the sea-air
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interface. Boundary-Layer Meteorology 1 (3):368-379
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Lan Y-Y, Tsuang B-J, Tu C-Y, Wu T-Y, Chen Y-L, Hsieh C-I (2010) Observation
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FigureA1. Left figure is the schema of the surface effective thickness ℎ𝑒 of an ideal
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surface with depth d. 𝑇0 is the skin temperature and 𝑇1 is the averaged temperature
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of the uppermost layer of the water with a thickness of ℎ1 . Note that the shaded area
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denotes the energy stored from the surface to depth d, which is close to the
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rectangular area. The mean temperature thus calculated from the rectangular area, the
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open circle beneath 𝑇0 , is representative of the skin temperature. 𝐺𝑖,𝑗 denotes the
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energy flux between layer i and j. Right figure is the schematic of the conventional
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discretization with a thickness of ℎ1 for the uppermost layer, representing only the
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bulk SST.
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