9. Auer, S. J., 1987: Five-Year Climatological Survey of the Gulf Stream System and. Its Associated .... obtained from extensive numerical experiments using a 2-D model of ..... using the CS approach for the eight consecutive days starting on.
U.S. DEPARTMENT OF COMMERCE NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION NATIONAL WEATHER SERVICE
NATIONAL METEOROLOGICAL CENTER
OFFICE NOTE 395
DYNAMIC COUPLING BETWEEN THIE NMC GLOBAL ATMOSPHERE AND SPECTRAL WAVE MODELS
D. Chalikov, D. Esteva, M. Iredell, and P. Long
August 1993
This is an unreviewed manuscript, primarily intended for informal exchange of information among NMC staff members
U.S. DEPARTMENT OF COMMERCE NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION NATIONAL WEATHER SERVICE
TECHNICAL NOTE*
DYNAMIC COUPLING BETWEEN THE NMC GLOBAL ATMOSPHERE AND SPECTRAL WAVE MODELS
DMITRY CHALIKOV, DINORAH ESTEVA, MARK IREDELL, AND PAUL LONG
AUGUST 1993
THIS IS AN UNREVIEWED MANUSCRIPT, PRIMARILY INTENDED FOR INFORMAL EXCHANGE OF INFORMATION *OPC CONTRIBUTION NO. 78 NMC OFFICE NOTE NO. 395
OPC CONTRIBUTIONS No. 1.
National
No. 2.
Development of Forecast Guidance for Santa Ana Conditions.
Burroughs, L. D., 1986: Weather
Digest.
Vol. 12 No
1, 8pp.
Wright, 1986: Lake Erie Wave Richardson, W. S., D. J. Schwab, Y. Y. Chao, and D. M. -- Comparison and Height Forecasts Generated by Empirical and Dynamical Methods Verification. Technical Note, 23pp.
No. 3.
Surface Lows Over the Auer, S. J., 1986: Determination of Errors in LFM Forecasts 313, 17pp. No. Note Northwest Atlantic Ocean. Technical Note/NMC Office
No. 4.
Rao, D. B., S. D. Steenrod,
and B. V.
Sanchez,
1987:
Total Flow from A Given Sea Surface Topography. l9pp. No. 5.
No. 6.
A Method
of Calculating
the
NASA Technical Memorandum 87799.,
Oceanographic Products Feit, D. M., 1986: Compendium of Marine Meteorological and NMC 68, 93pp. NWS Memorandum Technical of the Ocean Products Center. NOAA and ECMWF Numerical Model Auer, S. J., 1986: A Comparison of the LFM, Spectral, Technical Note/NMC Season. Cool One During Forecasts of Deepening Oceanic Cyclones Office Note No. 312, 20pp.
No. 7.
Regions. Technical Burroughs, L. D., 1987: Development of Open Fog Forecasting Note/NMC Office Note, No. 323., 36pp.
No. 8.
from Satellite Measurements Yu, T. W., 1987: A Technique of Deducing Wind Direction of Wind Speed. Monthly Weather Review. 115, 1929-1939.
No. 9.
Gulf Stream System and Auer, S. J., 1987: Five-Year Climatological Survey of the 11,709-11,726. 92, Its Associated Rings. Journal of Geophvsical Research.
Affected by Currents and Bottom No. 10. Chao, Y. Y., 1987: Forecasting Wave Conditions Topography. Technical Note, llpp. No. 11. Esteva, D. C., 1987: Technical Note, 4pp.
The Editing and Averaging of Altimeter Wave and Wind Data.
for Alaskan Waters. No. 12. Feit, D. M., 1987: Forecasting Superstructure Icing Weather Digest. 12, 5-10. No. 13.
Sanchez,
B. V., D. B. Rao,
and Indian Oceans.
S. D. Steenrod,
1987:
Tidal
Estimation
National
in the Atlantic
Marine Geodesy, 10, 309-350.
to T.W. Yu, and D.M. Feit 1988: Performance of Techniques Used No. 14. Gemmill, W.H., No. 330, 34pp. Derive Ocean Surface Winds. Technical Note/NMC Office Note Statistics of Techniques No. 15. Gemmill, W.H., T.W. Yu, and D.M. Feit 1987: Performance Workshop Proceedings Preprint. Used to Determine Ocean Surface Winds. Conference Scotia., 234-243. Nova Halifax. AES/CMOS 2nd Workshop of Operational Meteorology, A Method for Determining Equivalent Depths of the Atmospheric No. 16. Yu, T.W., 1988: ofGeophysical Research, 93, 3655-3661. Boundary Layer Over the Oceans. Journal Layer Heights Over the Oceans. No. 17. Yu, T.W., 1987: Analysis of the Atmospheric Mixed Workshop of Operational Conference Preprint. Workshop Proceedings AES/CMOS 2nd 425-432. 2, Scotia, Meteorology. Halifax, Nova An Operational Forecast System for Superstructure Icing. No. 18. Feit, D. M., 1987: the CoastalZone. 4pp. Proceedings Fourth Conference Meteorology and Oceanography of
DYNAMIC COUPLING BETWEEN THE NMC GLOBAL ATMOSPHERE AND SPECTRAL WAVE MODELS D. Chalikov, D. Esteva, M. Iredell, and P.Long National Meteorological Center, Washington, D. C. ABSTRACT small-scale of methods parameterization and theory The Two discussed. are ocean-atmosphere dynamical interaction briefly are problem the to different theoretical approaches One approach follows the one-dimensional theory presented. proposed by Chalikov and Belevich and the other is Janssen's Both approaches extension of the Miles' instability theory. marine boundary the of portion lower provide for coupling the wavy sea underlying the with layer, layer, the wave boundary approach each of simplifications applying of surface. The results and atmosphere global Center Meteorological to couple the National effects, The compared. and illustrated are models wave spectral although small, are noticeable especially in areas of high winds. 1.
INTRODUCTION
The local thermodynamic interaction between the ocean and the atmosphere is one of the primary mechanisms affecting the the of accuracy the Thus, system. ocean-atmosphere parameterization of this interaction determines to a considerable extent the quality of weather forecasts and climate modeling. Existing approaches to parameterize this microscale interaction do not take into account several mechanisms which influence both media. In this paper only the dynamical interaction is discussed, that is, the momentum exchange. However, the approach presented may be extended to take into account the sensible and latent heat exchanges and the effects of stratification. A major concern in the theory of the planetary boundary layer is the establishment of an appropriate relation between the turbulent stress T (bold letters denote vector quantities), and the horizontal wind velocity vector, u = (u,v) at an arbitrary height, z. It is generally accepted that this relation takes the form
T = Pa Cz I U I U
(1.1)
where pa is the air density, and Cz is the drag coefficient at height
z.
The value of C over land for neutral stratification depends upon the morphological surface characteristics, usually specified by the roughness parameter or roughness height z0, which is related to Cz by 1
Cz =
[ K/(ln(z/zo))]
2
(1.2)
,
where K is von Karman's constant. the The roughness height over the ocean is more complex due tofor a z estimate to o customary is It dynamics of the sea surface. (1955): Charnock by suggested formula the wavy sea surface with , u2 zO/g= m
(1.3)
where g is the acceleration og gravity, u, the friction velocity, been and m is an empirical coefficient whose experimental value has though (Garratt, 1977). Even found to vary from 0.01 to 0.05 a representative scale provide to shown been has expression (1.3) the for the roughness height over the ocean, the large scatter ofas a only empirical data indicates that it should be considered explained part in is qualitative relationship. The large scatter and by inaccuracies in the experimental technique, nonstationarity Scatter inhomogeneity of the flow, and density stratification. from may also be due to systematic deviations of the wind profile fluxes. momentum logarithmic which are caused by wave-produced sea Finally, (1.3) appears to be valid only for a fully-developed in the absence of swell.
(wave That the roughness length over the ocean depends on sea state Janssen (1968), Kitaigorodski (1967), age) was suggested by Stewart (1982), and Donelan (1982). The nondimensional wave age g is defined as g = c/u, , where cp is the phase speed of the dominant wind wave. The variability of the wave field introduces inhomogeneities in the surface stress over the ocean which amplifies the vertical motions at the upper levels of the marine boundary layer. Thus, the sea state may very well affect weather evolution and climate. The coupling between the wind and the waves was also considered by Jeffreys (1925), Miles (1957, 1960), Phillips (1957), Fabrikant (1976),
Chalikov
(1976,
1978,
1986),
Janssen
(1982,
1989),
and
Janssen extended Miles' shear-flow Janssen et al. (1989). of the Orr-Sommerfeld-Rayleigh solution numerical mechanism by equation with a diffusion-like a with tandem (OSR) equation in diffusion coefficient dependent on the wave spectrum and solutions to the OSR equation (Orr, 1907; Sommerfeld, 1980; Rayleigh, 1880). Expressing Phillips' constant as a power law with strong dependence on wave age which fits observed field wave spectra Janssen et al. (1984) found a strong coupling between the wind and waves for young (i.e. small wave age) wind waves, and a much weaker coupling for older waves. Janssen's approach was applied by Weber et al. (1992) to couple the wave model WAM (WAMDIG, 1988), to a general 2
atmospheric circulation model (ECHAM2, Roeckner et al., 1989). In the present study, a simplified form of the Janssen analysis is used which approximates the decrease in roughness length with wave age as found by Janssen. some approach has (1992) argued that Janssen's Chalikov difficiencies that precludes its confident use to model the air-sea In particular, Janssen's reduction of the momentum transfer. air-sea momentum transfer to a diffusion-like equation amounts to a diffusion equation with an advection term proportional to the vertical derivative of the diffusion coefficient. It is unlikely such an be modeled by that the momentum transfer may advection/diffusion process (Chalikov, 1992). 2.
WAVE BOUNDARY LAYER MODEL:
CHALIKOV'S APPROACH
A summary of the wave boundary layer model proposed by Chalikov and Belevich (1992) is presented in this section. Consider the bottom portion of the marine atmospheric boundary layer, the wave boundary layer (WBL), to be a nonstationary layer with a structure approximately governed by the equation:
a (T + at azau
(2.1)
where z is a vertical coordinate which at small heights will be considered as a surface-following coordinate, (Chalikov and are Belevich, 1992); u is the velocity vector; T and i turbulence to due the vertical fluxes of momentum respectively and to wave-induced perturbations. T may be parameterized as T
M~~~
~(2.2) auz
At a nonwhere K is the coefficient of turbulent viscosity. dimensional height ~ = z/Xa, the wave induced momentum flux may be expressed as
VA9 lr T= =@gS
CA,E.)dd2. ( ( 8) P ( a C,)F('t al)
9(2.3)
where pw is the water density, g is the acceleration of gravity, The function F(,aICAe) describes the and k is the wavenumber. variation of wave-induced momentum flux for each vertical nondimensional spectral frequency (a, 0a
=
6 I Ul I
e(24) COSO ~~(2.4) coS lu~~l
where uA is the wind velocity at a height z = Xa:
3
= ~a
2 2 g (D2 I C O s 0 I
(2.5)
is an apparent wavelength, and C is the drag coefficient at height I. The upper limit of integration or is the highest frequency for which the wave spectrum S is known. The function P is the nondimensional wind-wave growth parameter that is a function
of
for F and X are
Approximations
a and C A .
given in Chalikov and Belevich (1992). These approximations were obtained from extensive numerical experiments using a 2-D model of the statistical structure of the WBL (Chalikov, 1978; 1986; The wave Chalikov and Makin, 1991; Burgers and Makin, 1992). as defined is growth parameter (W) =
Pa
X ())
(2.6)
S( )
p ,,,
by d(e)is the energy flux from the wind to the waves divided as be approximated P may g(U) is the~~~~~~ the air density. where where
(-alt3Sa-a 2
·
was
-1
I
a 3t13
(a4 Wa - as)
-a6
ae(1
, I
(a 4 t5
- as)
~~2
>( 2a )
a
, lo a e (all
a7 va - as a9 (
32
a-1l) 2 + a1 0
ti5 ,
(2.7)
2)
> Q2
/
In (2.7) a1 through a10 and E1 and E2, are parameters depending on C and are given by = 1.075 + 75CA, a, = 0.25 + 395CI, a2 = 0.35 + 150CI, a4 = 0.30 + 300C1 , a9 = 0.35 + 240CI,
n2=
a10 = -0.06 + 470Cr,
a8
n
ao = 0.25
1.2 + 300Cx
as = (ao-a 4+a5) 2-a1l)/(ao+a a5 =
a 4 (1
(2.8)
a 6 = ao (1-a9 ) a7 =
a 52/a4·
4
(a9 (n
= 1 a7
2
-1)
2
+a10 )/ (nz-fl 1 )
For
frequencies
Ila.1 2
2
the
parameter
6
depends
&a2 and,
on
therefore, on the square of the wind speed. This dependence is confirmed by the experimental data of Hsiao and Shemdin (1983), and is more representative than the linear dependence derived from the field data of Snyder et al. (1981). High frequency waves are nearly stationary relative to the wind; thus, their drag in turbulent flow is likely to have a quadratic dependence on wind The speed, as is the case for stationary roughness elements. linear dependence of P on &a exists only for .a e (l , n2) . For & < -1, P becomes negative. This case corresponds to waves traveling against the wind, and suggests that in this case the waves transfer This is referred to as the inverse Miles' energy to the wind. mechanism. The use of u, and C; above eliminates the arbitrariness reduces the number of in choosing a reference level. It also governing parameters and is physically more meaningful because the thickness of the WBL decreases with frequency. The approximations (2.7) and (2.8) are also confirmed by Plant's (1982) data, including the quadratic dependence
The considerable scatter of additional dependence on C.
of P on 1a for lIaI > 2 (Fig. 1).
Plant's data may
be
due
to
the
Construction of a 1-D model of the WBL requires a definition of the vertical distribution of the wave-induced momentum flux T in (2.1). For a monochromatic wave traveling at an angle 0 to the wind, this is
= To F(u al CA,),
(2.9)
where To is the wave-induced momentum at the surface. The form of the scalar function F was investigated in numerical experiments with a 2-D model (Chalikov and Makin, 1991). An approximation based on the work of Makin (1989) is
F=
(2.10)
]e10
[1 -
where EO is a function of the drag coefficient C,
O = 0.1 + 60C x . Thus,
in practice
F
is independent
of &a
The
wave
frequency
influences the wave momentum flux at the surface, but not its vertical distribution. In the case of a multi-mode wave surface, it is assumed that T is the superposition of the fluxes due to all spectral components. The coefficient I, in (2.2) may be computed from
KM= Kz( e/c) 1/2
5
(2.11)
the turbulent kinetic where c1 is a constant equal to 4.6, and energy e may be computed from ae
at
au
= [KM
] a.
+
3
az
az
(2.12)
Kz
kinetic energy is In (2.12) the effect of diffusion of turbulent dot denotes the scalar not included since it is usually small; the product. general approach to be The model formulated above illustrates the system. The time and space used when modelling the WBL-wave field by a spectral wave evolution of the wave field may be simulatedaccomplished through is model. The coupling of the two systemsWBL and wave models. The the between the exchange of information wave model the spectral WBL model computes and transfers to the wave model computes the energy density input to the waves. The space and transfers the evolution of the wave spectrum in time and Equations (2.1) through (2.12) "sea state" to the WBL model. of the atmosphereconstitute a 3-D model describing the evolution connection between the wave coupling. These equations establish lowest layer of the the turbulent stress and wind in the are then used in the atmosphere. The updated stress components next time step. be estimated from The time scale Ts for the WBL processes may TS = h/(AC
z
(2.13)
U),
AC is the change in the where h is the thickness of the layer, and thez waves. Since ACz is drag coefficient due to the influence of 10 '3, T_ is on magnitude as Cz, that is, AC z of the same order of 3 a 3-D atmosphere model is the order of 3 x 10 s. The time step for it is reasonable to use on the same order of magnitude; therefore, the stationary version of the WBL equations: a
az
au
[Kmaz
+
az
(T+)
*
au
=
(2.14)
0
KM
(KZ)4
=
(2.15)
(Kz)
Equation (2.14) may be written in the form: T + T = Th ,
(2.16)
which equals the stress where Th is the constant of integration, written: vector above the WBL. Then (2.15) may be 6
Tba
az
( (KZ) 4
(2.17)
0
which may be rewritten as (KZ) 4/ 3 (h
aU ) 1/3 az
(2.18)
U
Taz
Since Th is unknown, two boundary conditions are necessary to solve (2.18). The first is the prescribed velocity vector at height h, z = h: U = U. 1
(2.19)
The second is the expression for the tangential stress vector at the water surface,
pa P a Cr |U X
KMLZ r K Z Zr: m
I U.:Z.o
(2.20)
The lower boundary height Zr is prescribed by the cut-off frequency Or:
2 7cg
(2.21)
=
z
It is assumed that the lowest part of the WBL is responsible for the creation of the local tangential stress due to waves with frequencies ( > or. Waves with frequencies X < er give rise to additional form drag which is taken into account in the integral (2.3). Assuming the Phillips' spectrum is valid for frequencies greater than r: (0 >
r: S((O)
= ag
2
W-
5
(2.22)
and that the local roughness 0 due to these waves is determined by their height, Chalikov and Belevich (1992) estimated 2
1-/2
c
°
g9-
(2.23)
with X = 0.1. Averaging the wave spectrum at o=or over all angles:
S
)'
,) , f S(2r
(2.24)
and equating this average to the Phillips' spectrum for X = or 7
S(()
= S(
r)(
(2.25)
)
(2.23) takes the form 2
= X
2 [S (G r) ] 1/2 Xs/2
(2.26)
The local drag coefficient is then given by
Cr = [K/(ln(zr/4)
(2.27)
)
An effective roughness parameter may then be found by combining An approximation to this (2.26) and the solution of (2.3). roughness
Equation
(2.18), with
formulas
(2.3) -
in (2.31) below.
is given
effective
boundary conditions
(2.5), (2.7), (2.8), (2.10),
(2.19), (2.20), and (2.21), (2.26), and
(2.27) formulate the parameterization of the WBL for use in coupling the atmosphere-wind wave system. Note that the integral in (2.3) depends on the vertical distribution of the wind u(z) and the drag coefficient Cz, which by definition is
CZ=
T
/ I u(z)
12
(2.28)
For the numerical solution of (2.18) it is necessary to evaluate (2.3) by iteration. The values for Cz and u(z) may be obtained from a linear interpolation of the logarithmic scale ln(z/zr) 1n= which, in practice, is convenient to use in the numerical solution of (2.18).
The approach presented above, although not overly complicated, takes a considerable amount of computer time. In the preliminary experiments discussed in this work, several simplifications were adopted: first, it was assumed that the spectra generated by the wave model includes the locally generated wind waves, which may be approximated by the JONSWAP spectrum (WAMDI group, 1988). The rest of the spectrum is considered to be swell and assumed not to This assumption, contribute to the wave-induced momentum flux. contribution swell the since differences large introduce should not the small to due small be should momentum to the wave induced steepness of these waves. 8
It was assumed that the spectral peak in the spectra generated by the wave model closest to the local wind corresponded to the The frequency of this peak was locally generated wind sea. determined by minimizing the expression
(kx - kxr) 2 + (kyi - kyr)
(2.29)
where kxi,k.n are the x,y components of the wave number for spectral peak i, and kxr, kyr are those of the wave number of the shortest represented wave, which is assumed to be in the direction of the wind. Minimizing (2.29) is equivalent to minimizing
4+ 4-
2)2( rCOS(i
Ow)
(2.30)
in frequency-direction space. m) and 0. are the frequencies and directions of the different peaks, 0 w is the wind direction, and r is the frequency of the shortest wave represented in the spectrum. Once the frequency of the local wind wave is selected, its corresponding phase speed is used to determine the inverse wave age, and the effective roughness height z0 is computed from z0 = (u 2 /g)exp(-11.35+0.187R+(0.84+0.065R)
(u/cp))
with R = ln(u2 /(gz)). Equations (2.7) and (2.8), were not introduced. Only the largescale coupling between the atmosphere and the waves has been, considered, hence equations (2.7) and (2.8), were not introduced. 3.
SHEAR FLOW MECHANISM:
JANSSEN'S APPROACH
The theory of wind generated gravity waves as presented by Miles (1957, 1960, 1976) is similar to the theory of the resonant interaction of plasma waves with particles (Landau and Lifshitz, 1954; Fabrikant, 1976). Surface waves in plane-parallel flow are generated with phase velocities equal to the wind velocity at some "critical" (or resonant) level, zc. Depending upon the influence of the wave age on the Phillips' parameter in the wave spectrum, the wave-induced stress can be a significant fraction of the total stress (Janssen et al., 1989). For waves traveling along the mean horizontal wind u, and subject to certain conditions, Janssen (1982) has shown that the wind-wave interaction may be modeled by a diffusion-like coefficient, D,(k,z),
Dw(k,z) =
2k C - Vg
9
¢(k)
Ix(k,z) 12.
(3.1)
be dropped In the treatment that follows, vector notation will mean wind. the with aligned are since it is assumed that the waves satisfies wind mean the stress, Together with turbulent and viscous the equation,
au/at
=
a
2u
1 aT pau p azz
+
az2
+
Dw(kz) -
-
(3.2)
(3.2)
be modeled in which T is the turbulent stress which is assumed to by a mixing length theory. the group In (3.1, 3.2), vg, 0 (k), va, D (k,z), and x(k,z) denote: phase speed c,
velocity of a surface wave with wave number k and the the spectral amplitude, the molecular viscosity of air, and diffusion-like coefficient that models the air-sea interaction, the normalized solution of the OSR equation, respectively. The Orr-Sommerfeld equation is a 4th-order ordinary differential equation: 2 ! xiv_ 2k2 X" + k4 X = ikRe[(u - c)(X - k X)-u!x]
If Re is taken to be infinite,
in which Re is the Reynolds number. then U
2 2 - k2X dz
which is the OSR equation. for (3.4) are
U- C
X
=
0 ,
(3.4)
For the MBL, the boundary conditions
X (z
x
(3.3)
0) = 1
-
(z
co) =
-*
0
(3.5)
to a The function X(z,k) arises from a perturbation solution Let the nondimensional follows. flow as plane-parallel terms of a mean and a in expressed be equation Navier-Stokes perturbed flow:
p = P +p/ U = U +u W =
10
W/
(3.6)
az aX
w/ =
(3.7)
ax
then
aUaua j.a* ap +
a2 + u, axaz
azat
___+
~ X-t
axat
+U
az
ax
ax
o az
Re
apl 12 p + Va a x+a
ax 2
(3.8)
ax
Re
az
Using the separation
where primes indicate perturbed quantities. of variables technique with trial solutions
(x,z,t) = X(z)exp[ik(x-ct)] p/(x,z,t) results
in (3.10), and
(3.9)
= I[(z) exp[ik(x-ct)]
(3.11) below.
X = -ikI + -ik dU ik(U-c)dA dz dz k2(c-U)X-
d= M dZ
(
Re dz 3
ik d2X2 Re dz -
k2)
-
k d
dz
(310)
(3.10)
(3.11)
Equations (3.10) and (3.11) are a coupled system from which 1(z) may be eliminated to get (3.3) and (3.4). The Rayleigh equation, (3.4), has a singularity at the critical Since (3.6) - (3.11) are nondimensional, X(z,k) is height, z . The steps used by Janssen to solve the preceding normalized. equations and estimate the wind-wave interaction are given schematically in Appendix A. For a following
wave
spectrum
O(k,O,t),
11
Janssen
(1991) gives
the
D (k,z)=
f
dO
IX1
2
(k,0, t)cos
2
0
(3.12)
in which 0 is the angle between the wind and the spectral wave component. A simplified version of Janssen's approach is used in this work. The simplification consists in deriving a generalized Charnock's constant as a function of wave age using Janssen's analysis. This generalized Charnock formula is then used to compute the roughness height z0 corresponding to the wave age M as determined from the frequency fp and wave number kp of the "sea" spectral peak:
= C/u*,
with c
=
g/kp.
This leads to a wave age dependent drag coefficient Cz. This roughness height is passed from the wave model to the atmosphere model which then generates the winds which will drive the wave model for the next time step. In this fashion a two-way coupling between the wave and atmosphere models is achieved. of The generalized Charnock relation is derived from the familyFig. adapted
curves
from Fig. 3 of Janssen
(1989)
(shown here
in
These curves represent typical nondimensional velocity 2). profiles with and without wave perturbed flow. In the figure the the versus plotted is u(z)/u, velocity nondimensional 2 and straight nearly are curves The . nondimensional height zg/u, 2 < 1 the perturbed curves deviate z 2 zg/u, parallel for zg/u > 1. For boundary condition (A.3), cross the from log-linear, and, given 2 u/u* = 0 intercept at zg/u* = m = 0.0144. The extrapolation is not shown in the figure. Fig.
7
of
Janssen
(1989),
shown
here
as
Fig.
3,
shows
the
dependence of Cz at a 10 m height on wave age for two values of u, and the (0.3 m/s and 0.7 m/s). Curves are shown for both the -2/3 A). It Appendix (see parameter Phillips -3/2 power laws for the wave with C, of variation considerable is can be seen that there age for the -3/2 power law. This power law was used in obtaining the curves in Janssen's Fig. 3. The simplification used in this work was suggested by Fig. 2. For zg/u*2 > 1 the curves are merely displaced from the straight line This representing the uncoupled wave-atmosphere situation. suggests that a general equation for the family of curves may be written as:
12
(3.13)
u(z)/u* = A ln(zg/u2 ) + B(p)
where A is the slope of the family of curves, and B(A) depends on wave age only. An average value of A is 2.46. Equation
(3.13) is rewritten
as
(3.14)
[zg/D(V)u~]
u(z) /u* =A ln in which B(p)
= -A ln[D(p)]
(3.15)
,
thus,
C
1/ 2-
= u(z)/u. = A ln[ (1/C z) (zg/D(p) u 2)]
(3.16)
In similar fashion the Charnock relation (1.3) may be written 2 Cz 1/ 2 = K -1 ln[(1/Cz ) (zg/m u )]
.
(3.17)
Comparing (3.16) and (3.17) suggests: A =
K
(3.18)
1,
and
(3.19)
D(A) = a generalized Charnock "constant" which approaches m for large values of g.
From (3.16) through (3.18), the generalized Charnock "constant" is:
D(>) = (zg/u2 ) [C 1 exp(-0.407CC /2·)] .
(3.20)
Computing the D(p) for different values of wave ages yields A
5I _
10
1
15
20
_25
D(,!) 0.210! 0.727! 0.414! 0.0328! 0.02771 A rational fraction that approximates D(p) for A > 3 is given by
D(p) = (-0.153 - 0.00245[)/(1 13
- 0.358p)
(3.21)
This approximation for D(y) is used to generalize the Charnock relation: (u2 /g) = D(p) ([) (,/g).zo = D
4.
THE NMC ATMOSPHERE AND WAVE MODELS
are Brief descriptions of the NMC atmosphere and ocean wave modelseach of independently run models these given below. Operationally the other, with the wave model using the output wind fields from model atmosphere model as the sole input. The operational wave wave runs twice daily at the 0000 and 1200 UTC cycles. A so-called that is, analysis is generated by running a 12-hour wave hindcast,analysis" This "wave using the final analysis wind fields. 72-hour forecast. The a for field wave initialization the provides winds input wind field to the wave model is the lowest sigma layer For profile. wind reduced to a 10-m height assuming a logarithmic u* the take to modified the coupling experiments the wave model was once done were runs The field directly from the atmosphere model. fields daily, for the 0000 UTC cycle only, thus initialization wave running by done was hindcast The resulted from a 24 hour hindcast. 24 the NMC Global Data Assimilation System (GDAS) for the preceding forecasts. the for as technique hours with the same coupling 4.1
The NMC atmosphere model
for The operational NMC Medium Range Forecast Model (MRF) is used 10-day the GDAS, the Aviation (3 day) forecast, and for the in forecast runs. The MRF is a spectral model and is described A some detail in Kanamitsu (1989), and Kanamitsu et al. (1991). few of its properties are described here. The MRF forecast variables are vorticity, divergence, virtual temperature, specific humidity, and surface pressure. The model variables in the horizontal are represented by spherical harmonics. vertical and a horizontal The model has 18 levels in the resolution of triangular truncation 126 (T126). This truncation corresponds to a horizontal spatial resolution of approximately 105 However, in order to save computer time, the coupled km. a atmosphere-wave experiments described here were performed with spatial a to corresponds triangular truncation of 62 (T62), which resolution of approximately 210 km. is The lowest layer of the atmosphere model, the surface layer, surface. the above well defined with a height of 5 hPa (about 40 m) The physics in this layer is governed by surface layer MoninObukhov similarity theory and improved land surface evaporation. the The atmospheric boundary layer (ABL) in the MRF consists of influenced bottom surface layer and those layers above it which are 14
by the turbulent transfer of momentum, heat, and moisture from the surface layer. During vigorous daytime turbulent convection the ABL is typically 1 - 2 km thick. However, during extremely stable conditions, such as may occur at night, turbulence may become so weak and irregular that it may not be possible to determine accurately the height of the ABL. For these conditions, the ABL The MRF makes no may recede into the surface boundary layer. occurrence. this for provision The MRF computes the surface turbulent fluxes of momentum, sensible The heat fluxes are required for the heat, and latent heat. equations that predict the air-ground balance solution of the heat heat balance calculation is omitted This interface temperature. is important in predicting the it although over the oceans fluxes of momentum and heat are surface The MBL. the evolution of the diffusion equations which for conditions boundary as used simulate the turbulent transfer of these quantities above the surface layer. The calculation of surface fluxes in the MRF depends on the nondimensional variable zs/L, where zs is the height of the surface layer and L is the Monin-Obukhov length, the sign and magnitude of which determines the stratification/stability of the surface layer. The vertical profiles of (virtual) potential temperature e, specific humidity q, and wind speed in the surface layer are given by
a-
az
az =k
az
z
K
Kz
au u. auUau az
Kz
-~~~
WT(P/L)
(4.1)
P q(z,L)
(4.2)
u(z/L)
-
kH
~~~~~(4.3)
(4.3)
(U2 Ou~
with L=
KgO.
0*, q*, and u* are the scaling factors for heat, moisture, and momentum. The functions (T qu are the Obukhov functions which have been determined empirically.
15
Integration of the Obukhov functions leads to
zs/L = (Ft/FT)Rb,
(4.5)
in which Rb is the bulk Richardson number, gzsA O b-
oRU 2
and FTUq are the integrals of equations (4.1) - (4.3), FT U'q =
z
dz
Z
(PT,uq(Z/L)
(4.6)
In these equations, A 8 is the difference in virtual potential temperature across the surface layer, 0 is the layer-average virtual potential temperature in the surface layer, us is the wind roughness height. speed at the top of the layer, zs, and zo is the that the roughness assumed is it MRF the of In the current version Over the equal. are moisture and heights for momentum, heat, by proposed form the as taken is z ocean the roughness height o (m=0.014). constant Charnock the is m which in Charnock: (eq. 1.3), u., Iterative solution of (4.5) using (4.6) yields values for zs/L, are 0., and q,, and from these the surface fluxes FM', FT', Fq' computed: U2 =2) FM (4.7) F;M= -P aU* = -PaCm(zs,L) us
=
-p a Cp CT(Zs,L) us A 0
(4.8)
Pa S u* q. =
Pa S Cq(ZsiL) us A q
(4.9)
FT = -P a
Fq =
Cp U* e*
C., CT, and Cq are the drag (or transfer) coefficients for momentum, heat, and moisture respectively; 9 is the latent heat of C is the specific heat of air, and Aq is the vaporization, difference in specific humidity between z and zo. Empirical data The MRF computes the transfer suggest that CT = Cq ~ CM. coefficients from
16
U,/2'
(~M) M~us
CT = Cq =u
U, O,
A
=
K2
FMf K2
U, q9 =
q
uA
(4.10)
12
FF
>-{}g 4 *; 7;-il.;-; .. .X. * 3. J
. :
60E
9U0E
r_
1 lU I bUE IUU MERN= 0.010
Figure lla. 53
Figure l1b. 54
SWH DIFF. [m] between coupled & control runs 96h FCST VALID
@
2/ 3/92
0Z
60.
30.
0.
-100.
-160.
-1.25
-1.00
-0.75
CONTOURFROH -1.7500
-0.50 TO
1.2500
0.00
-0.25
0.25
CONTOURINTERVALOF 0.25000
Figure 12a. 55
0.50 PT13.3)-
0.75 0.00000E00
1.00
control
SWH DIFF.[m) between coupled & control 96h FCST VALID
@
2/ 3/92
runs runs
0Z
60.
30.
9.
20.
-35.
-9s.
i
-1.25
-1.00
-0.75
CONTOURFROM -1.7500
-0.50 TO
1.2500
0.00
-0.25
0.25
CONTOURINTERVALOF 0.25000
Figure 12b. 56
0.50 PTt3.3)=
0.75
1.00
0.09090E.00 -
.
SPECTRA - 92020300 CONTROL RUN: DIRECTIONAL AND ONE DIMENSIONALfrom north(up) Frequency interval= 0.039 - 0.308. Directions DIR= LAT/LON= 45.0/-130.0. SWH= 8.3(m). U.STARt10= 7.501m/s).
19A
9,
F
I
