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Bell & Howell Information and Learning
300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA
800-521-0600
by
K o n stan tin Zahariev
B.Sc., U niversity of Sofia, 1992
A D issertation S u b m itted in P a rtia l Fulfillment of th e Requirem ents for th e Degree of
D O C TO R O F PH ILO SO PH Y in the School of E a rth an d Ocean Sciences
We accept this d issertation as conforming to th e required standard
Dr. Chris G arrett
Supervisor (School of E arth and O cean Sciences)
D r. Ken Denman
Depart^ rental M em ber (School of E a rth and Ocean Sciences)
Dr. Andrew Weaver
D epartm ental M ember (School of E a rth and Ocean Sciences)
D r. N orm M cFarlane
O utside M ember (C anadian C entre for C lim ate Modelling and Analysis)
D r. Eric D ’Asaro
E x tern al Exam iner (University of W ashington)
© K o n stan tin Zahariev, 1998 U niversity of V ictoria
All rights reserved. This dissertation m ay not be reproduced in whole or in p a rt, by photocopying or other m eans, w ithout th e permission of th e author.
Il
A b s tr a c t
T h e oceanic surface boundary layer is of great im portance and interest as
its dynam ics provides for the exchange of energy, m om entum , heat an d m a tte r be
tw een th e atm osphere and the ocean. It is crucial to have a thorough understanding
of physical processes th a t m ight have a significant influence on its properties zmd
variability. In this stu d y 1 consider several different facets of m ixed lay er/b o u n d ary
layer dynam ics.
One aspects concerns th e consequences of th e nonlinearity of th e equation
of s ta te in m ixed layer models. T he n onlinearity of the equation of s ta te gives rise to a te rm in th e averaged surface buoyancy flux which can be com parable in
m agnitude to o th er term s. Its m agnitude is shown to be proportional to the area
enclosed by th e seasonal cycle of sea-surface te m p e ra tu re T versus th e oceanic heat
content 1-L. T h e te rm always represents a buoyancy in p u t into the ocean and is
com pensated exactly by the buoyancy loss via cabbeling (densification on m ixing)
whenever th e m ixed layer entrains w ater w ith different properties from below.
A nother problem of interest is th e role of th e coherent w ind-induced vor
tices, com m only known as Langmuir circulation, in generating th e surface m ixed
layer. A sim ple param eterization of th e m ixing due to Langm uir circulation is
exam ined in the light of an oceanic d ataset. Some evidence for th e validity of th e
p aram eterizatio n is found, thus draw ing a tte n tio n to Langm uir’s assertion th a t
Langm uir circulation is one of the key physical processes in th e oceanic boundary layer.
T h e th ird aspect of surface boundary layer dynam ics explored is th e m ean
effect on m ixed layer entrain m en t of periodic vertical m ovem ent of isopycnals in
th e therm o d in e due to non-breaking in ternal waves (referred to as heaving). Sea
sonal m odel runs incorporating idealized heaving show th a t heaving can produce
significant seasonal differences in sea-surface te m p e ra tu re com pared to a reference
case w ithout heaving. It is inferred th a t by periodically stretch in g and com pressing
th e m ixed layer, heaving causes instabilities th a t result in additional en train m en t
of colder w ater from below. A heaving num ber is proposed, and two param e-
terizations of heaving for use in m ixed layer models are suggested.
Dr. Chris G a rre tt
Supervisor (School of E a rth and O cean Sciences)
Dr. Ken D en m an
D epart^ rental M em ber (School of E a rth and Ocean Sciences)
____________________________________________________ Dr. A ndrew Weaver
D ep artm en tal M em ber (School of E a rth and Ocean Sciences)
_________________________________________ Dr. N orm M cFarlane
O utside M em ber (C anadian C entre for C lim ate M odelling and Analysis)
Dr. Eric D ’A saro
IV
T able o f C o n ten ts
List of T a b l e s ... vii
List of Figures ... viii
A cknow ledgm ents... xv
D ed ic a tio n ... xvi
C hapter 1. I n tr o d u c tio n ... 1
1.1 M otivation and o u t l i n e ... 1
1.2 T h e oceanic surface boundaxy l a y e r ... 3
1.2.1 O v erv iew ... 3 1.2.2 F o rc in g ... 4 1.2.3 P ro c e sse s... 9 1.3 O b s e rv a tio n s ... 19 1.3.1 OWS P a p a ... 19 1.3.2 LOTUS ... 20 1.3.3 M ILE ... 20 1.3.4 P A T C H E X ... 21 1.3.5 GATE, phase I I I ... 21
C hapter 2. M ixed layer m o d e l s ... 23
2.2 T h e Price-W eller-Pinkel m o d e l... 26
2.3 T he Mellor-YcLmada Level 2 m o d e l... 31
2.4 O th er m o d e l s ... 33
C h ap ter 3. Noolinear e q u atio n of state and ap p aren t buoyancy flux . . . . 39
3.1 I n tr o d u c tio n ... 39
3.2 T h e o r y ... 41
3.2.1 R ep resen tatio n of th e term involving clQ ... 41
3.2.2 G eneral properties of th e seasonal cycle of versus "K 47 3.3 Seasonal cycle m odelling - a te st c a s e ... 50
3.4 C alculation of th e NES term from OWS P a p a d a t a ... 55
3.5 Discussion a n d c o n c lu s io n s ... 57
C h ap ter 4. Langm uir circulation and m ixed layer d e e p e n i n g ... 61
4.1 I n tr o d u c tio n ... 61
4.2 The LC criterion for ML d e e p e n i n g ... 63
4.3 Verification of th e LC criterion using LO TU S d a t a ... 66
4.4 Im plications for u p p er ocean r e s t r a t i f i c a t i o n ... 78
C h ap ter 5. Isopycnal heaving and its effect on en train m en t ... SO 5.1 I n tr o d u c tio n ... 80
5.2 Models an d sensitivities to th e background diffusivity . . . 81
5.2.1 A relationship betw een background diffusion and en tra in m en t 83
VI
5.3 Effects of isopycnai heaving on e n tra inm e n t ... 91
5.3.1 A model w ith isopycnal h e a v i n g ... 91
5.3.2 Seasonal runs w ith different heaving param eters . . . . 98 5.3.3 A detailed look a t a three-day period in th e su m m er . . 104
5.3.4 Heaving n u m b e r ... 110
5.4 P aram eterizing heaving for use in m ixed layer m o d e l s ... 115
5.4.1 Heaving as an equivalent 115
5.4.2 Heaving as a correction to th e dynam ic sta b ility criterion 119
5.5 S um m ary and conclusions ... 124
C h ap ter 6. O u t l o o k ... 126
VIII
L ist o f F igu res
1.1 Some of the m echanisms affecting the turbulence in the oceanic su r
face boundary layer: Langmuir circulation, breaking surface waves,
shear-induced turbulence, convection, breaking internal waves (re
drawn from T horpe (1985))... 10
1.2 A schem atic of th e nonlinear dependence of density of seaw ater on
tem perature. W arm w ater expands more th a n cool w ater contracts. If water w ith tem p eratu re T l, density Z)l, mixes with w ater w ith
tem perature T2 > T l , density D'2 < D l, in equal volumes, th e m ixture, while of tem p eratu re T m = { T l + T 2 ) / 2 ^ will have density
D { T m ) larger th an th e m ean density D m = ( D l + D 2 ) / 2 ... 12
1.3 A schematic representation of Langmuir circulation (redraw n from
Pollard (1977))... 14 1.4 Ten-day tim e series of isotherm s versus dep th an d tim e, taken from
th e LOTUS d ataset (Briscoe and Weller 1984; Bowers et al. 1986). The high-frequency variability in the isotherm s’ position in an d
below the seasonal therm ocline at 20 m etres d ep th is indicative of
3.1 A schem atic of m ixed layer deepening due to m echanical stirrin g an d associated entrainm ent. A well m ixed surface layer of d ep th
h, te m p e ra tu re T , and a te m p e ra tu re ju m p A T across its base
deepens, in tim e to a depth h + 6h, lowering its te m p e ra tu re by
6 T to T - S T ... 44
3.2 T h e annual cycle of versus 'H a t OW S Echo (35°N, 48°W ),
m odified from Gill an d Turner (1976)... 48
3.3 Idealized forcing w ith zero m ean an n u al heat flux. Top: insolation
Qg, to ta l heat loss (tbe sum of sensible, la te n t, an d longwave),
an d net h eat flux Q. Bottom : eastw ard wind stress ... 51 3.4 T h e annual cycle of T^ versus Ti from m odel o u tp u t, if th e NES is
excluded (upper solid fines), an d including it (dashed fines). T h e
pairs of fines for each case delineate th e diurnal ranges. T h e lower curve shows th e loop, with th e NES b u t when th e diu rn al cycle is
not resolved. T h e m inim um annual h e a t content is a free p a ra m e ter which does not en ter the calculations of th e NES term , an d is set
to zero. Each circle corresponds to th e m iddle of a m o n th ... 53
3.5 (top) T he daily range (i.e. daily m inim um and daily m ax im u m
values) of m ixed-layer depth h if th e NES is excluded (solid fines) an d including it (dashed fines); h is slightly larger on average in th e
NES case, (b o tto m ) T he daily range of sea-surface te m p e ra tu re T^
if th e NES is excluded. Its inclusion results in slightly lower average
3.6 T h e annual cycle of versus H for th e year 1972 from OWS P a p a
d ata. T he m inim um annual h e at co n ten t is set to zero. Each circle
corresponds to th e middle of a m o n th ... 58
4.1 A schem atic diagram showing th e interactio n betw een Langm uir circulation and pre-existing stratifica tio n . I: the general problem , II: startin g from a linear stratifica tio n , and III: sta rtin g from a two-layer stratification. Redrawn from LG97... 64
4.2 Time-series of th e difference betw een th e tem p eratu re at a d e p th of 0.6 m (considered a sea-surface tem p eratu re), and the te m p e r a tu re a t a d e p th of 5 m etres. T h e surface tem p eratu re is consis ten tly lower by about 0.06°C even a t night when the u pper ocean appeared well m ixed... 68
4.3 Top (right) panel: an exam ple of tem p e ra tu re contours an d cor responding estim ates of h (u p p er thick line) and (lower thick line); B ottom (left) panel: Forcing (wind speed and insolation) for th e same period... 69
4.4 Com parison of th e heat content estim a te s for the upper 51 m etres of ocean as derived from tem p eratu re profiles (shaded) and as im plied from the local surface heat fluxes (th ick black)... 71
4.5 T h e caption is on page 7 0 ... 74
4.6 T h e caption is on page 7 0 ... 75
4.7 T h e caption is on page 7 0 ... 76
5.1 T em p eratu re profiles after 50-day runs with, th e MY-2 m odel for different background diffusivity values, and for forcing w ith zero
m ean to ta l heat flux, constant eastw ard w ind stress of 0.1 P a. From
right to left a t th e surface: initial profile; 50th day final profiles for
ATyg = 0; 1 X 10 ^;5 X 10 ^... 87
5.2 Idealized forcing w ith zero m ean annual heat flux. Top: insolation
Q^, to ta l heat loss (th e sum of sensible, la te n t, and longwave),
and to ta l heat flux Çt- B ottom : eastw ard wind stress ... 89
5.3 Seasonal runs w ith MY-2 m odel for two different background diffu sivity values: lxlO ~^m ^s~^ (m iddle), and 5 X 10~“m^s~^ (b ottom ).
Com pare th e m axim um seasonal SST at th e end of sum m er with
th e reference (/f^g == 0 (top) case)... 90 5.4 Ten-day tim e series isopycnal plot of dep th versus tim e, from the
LOTUS d ataset. Note th e coherent heaving movement of isopyc
nals regardless of depth. Also shown is the estim ated m ixed layer
d e p th ... 92
5.5 Seasonal runs w ith the P W P m odel for three different background
diffusivity values: 0.7 x 10~^m^s~^ (1), 1 x 10 (2), and 1.5 X 10~^m^s~^ (3). Also shown for com parison is the reference
(ATbg = 0 (top) case)... 95 5.6 O ne-day (year-day 202) modified P W P m odel o u tp u t showing heav
X II
5.7 Seasonal runs w ith a modified P W P model for two different am plitudes of isopycnal heaving: = 2.0m (1), an d = 5.0m (2),
w ith a fixed wave period of 1.6 h. Those are co m p ared to the
reference (top) no-heaving case... 100
5.8 Mixed layer d ep th from a modified P W P m odel fo r two differ
ent am plitudes of isopycnal heaving: A^ = 2.0m (to p panel), and A^ = 5.0m (b o tto m panel), ivith a fixed wave p erio d of 1.6 h.
Those are com pared to th e m ixed layer dep th (daily range) for the
reference case... 102 5.9 Seasonal runs w ith a modified P W P model for th re e different pe
riods of isopycnal heaving: = 2.4 h (1), — 1.2 h (2), and = 0.4 h (3), w ith a fixed wave am plitude of A^. o f 2.0m . Those
are com pared to th e reference (top) no-heaving case... 103
5.10 Mixed layer d e p th from a modified PW P model for two different periods of isopycnal heaving: = 1.2 h (top p an el), an d =
0.4 h (b o tto m panel), w ith a fixed wave am p litu d e of 2.0 m . Those are com pared to the m ixed layer dep th (daily range) for the
reference case... 105
5.11 Contour plots of d ep th versus tim e, for days 200-202. Top panel:
m ean (over a heaving period) tem perature contours for th e refer ence case (no heaving). B ottom panel: m ean te m p e ra tu re contours
for a heaving case w ith an am plitude of 2 m etres a n d a period of
5.12 C ontour plot of depth versus tim e, for days 200-202, of th e m ean te m p e ra tu re difference (over a heaving period) betw een th e refer
ence case (no heaving) and a heaving case w ith an a m p litu d e of
2 m etres and a period of 1.6 hours. T h e plot shows th e u p p e r 20 m etres of m odel ocean. T he m ixed layer depth for th e reference
case is shown in w hite... 108
5.13 Seasonal runs w ith a modified P W P m odel for a heaving w ith =
2.0 m and — 1.6 A: original ru n with continuous heaving (1)
a n d a run w ith heaving sw itched off from 8:00am to 4:00pm each
day (2). T h e no-heaving case is also shown for reference... I l l
5.14 Values of for pairs of d ata-p o in ts with the sam e heaving n um ber R jj b u t different nondim ensional frequency Tq/ T ^ . P airs
of points are denoted by the sam e symbol. Pairs: “o” , = 1.64:
R ^ = 2.22; “A ” , R ^ = 2.82; R ^ = 4.47; R ^ = 7.69. 114
5.15 D ependence of th e scaled m axim um seasonal SST difference
th e heaving num ber R ^ (top panel), an d on th e scaled background diffusivity R ^ (bottom panel), w ith corresponding in terp o lated val
ues (dashed lines; in the top panel th e dashed line indicates a crude
piece-wise linear interpolation)... 116 5.16 Correspondence between th e scaled background diffusivity R ^ an d
th e heaving num ber Ry^ (top p an el), and equivalent background diffusivity (in m^s~^) as a function of R ^ (b o tto m p an el),
w ith corresponding interpolated values (dashed lines, in d icatin g a
XIV
5.17 D ependence of th e scaled m axim um seasonal SST difference on
th e critical Richardson num ber R b ^ (top panel), an d a correspon dence betw een th e stability ad ju stm en t factor a n d th e heaving
num ber (b o tto m panel), w ith corresponding in te rp o la te d val ues (dashed lines; in th e bo tto m panel the dashed line indicates a
crude piece-wise linear in terp o latio n )... 121 5.18 A com parison betw een seasonal H — curves for a heaving ru n
(black line) w ith = 3.0 and a diffusivity run w ith an equivalent
background diffusivity of = 0.8 x 10~^m^s~^ (top panel). T h e
b o tto m panel shows a com parison betw een th e sam e heaving run (black line) an d an adjusted critical R ichardson n u m b er ru n w ith
A ck n o w led g m en ts
I would like to thank my supervisor Dr. Chris G a rre tt for his guidance an d
generous su p p o rt. I also th ank m y external exam iner, D r. Eric D’Asaro, an d th e
m em bers of m y com m ittee, Dr. Ken D enm an, Dr. Andrew Weaver, and Dr. N orm M cFarlane, for th e ir support and for valuable com m ents and suggestions. I th a n k
m y fam ily for th e ir love and su pport. I would also like to th a n k the U niversity of
X V I
D e d ic a tio n
C h a p t e r 1
I n tr o d u c tio n
1 .1 M o t iv a t io n a n d o u t lin e
O ceanic an d atm ospheric variability fascinates w ith its com plexity on any tem p o ral and sp atial scale. T he oceanic surface boundary layer^ is of p artic u la r
im p o rtan ce an d in terest, as its dynam ics provides for the exchange of energy, m o m en tu m , h eat and m a tte r betw een th e atm osphere and th e ocean. Its seasonal
variability controls th e exchange of heat w ith the therm ocline. T h e properties an d dynam ics of th e m ixed layer influence chem ical and biological processes in th e
ocean: alm ost all biological a ctiv ity takes place in the u p p er layer of th e ocean.
M ixed layer dynam ics governs th e sea-surface tem p eratu re a n d th u s th e fluxes of
h eat, m oisture, an d gases to th e atm osphere; it is one of th e m o st im p o rta n t factors d eterm in in g clim ate.
Consequently, it is crucial to have a thorough u n d ersta n d in g of physical
processes th a t m ight have a signiflcant influence on the pro p erties an d variability
of th e surface boun d ary layer. In th is stu d y I consider several different facets of m ixed la y e r/b o u n d a ry layer dynam ics.
^The term s “oceanic surface boundary layer” , “surface boundary layer” , “surface m ixed layer” , and “m ixed layer” m ay be used interchangeably in tliis study. T h ey denote the su rface layer o f water o f quasi uniform tem perature, salinity, and buoyan cy/d en sity, but m ay also include th e underlying therm ocline d epending on co n tex t. In the la tter case, the term “boundary layer” is usually preferred.
1. In troduction
O ne of th e aspects I look at pertains to th e consequences of th e nonlinearity
of th e eq u atio n of s ta te in m ixed layer models. As noted by G arrett et al. (1993) for th e M editerraneaji Sea, th e nonlinearity of th e eq u atio n of state gives rise to an
additional te rm in the averaged surface buoyancy flux which can be com parable in m agnitude to th e o th er term s. T he term always represents a buoyancy input into
the ocean even when th e an n u al mean heat flux is zero, so a com pensating buoyancy
loss over th e seasonal cycle m ust occur, via cabbeling (densification on m ixing) whenever th e m ixed layer entrains water w ith different properties from below. A
sim ple representation of th e m agnitude of th is te rm is derived, relating it to th e the seasonal cycle of sea-surface tem perature versus th e oceanic heat content
and general properties of th e seasonal cycle of versus H are discussed^.
A n o th er problem in this area of interest is th e role of the coherent wind-
induced vortices, com m only known as Langm uir circulation, in establishing and
m aintaining th e surface m ixed layer. Following up work by Li and G arrett (1997) on a sim ple param eterizatio n of th e mixing due to L angm uir circulation from which
a Langm uir circulation criterion was derived for use in bulk m ixed layer models, 1 collaborated w ith the authors to verify th eir form ulation by directly exam ining an oceanic d a ta set, looking for m ixed layer deepening events th a t did not appear to be
caused by shear instability a t th e base of th e m ixed layer b u t were predicted w ith the new criterion. Im plications for seasonal an d d a y tim e ocean restratification are
also discussed".
^This reseajTch was published as Zahariev and Garrett (1997). ^This research was published as Li, Zahariev, and G arrett (1 9 9 5 ).
T h e th ird aspect of surface boundary layer dynam ics I explore is th e mean
effect on m ixed layer entrainm ent of periodic isopycnal up-down m ovem ent in the
therm ocline due to non-breaking internal waves (henceforth referred to as “heav ing” ). M odel o u tp u t, such as sea-surface tem perature, from one-dim ensional mixed
layer m odels shows th a t models are sensitive to th e vaiue of background diffusiv
ity iifyg used below th e mixed layer, in the therm ocline. It is thus im p o rtan t to estabhsh w h eth er represents m ixing processes in th e therm ocline or is a proxy
for processes o m itted from the models. One such process is isopycnal heaving. I
exam ine th e sensitivity of the m ean entrainm ent ra te to th e heaving a t th e base of the m ixed layer by considering a modification of th e P rice et al. (1986) m ixed layer model th a t allows for periodic isopycnal displacem ent superim posed on th e other
variabihty. Results from seasonal runs w ith this m odel and idealized forcing are
presented, along w ith estim ates of the equivalent background diffusivity. A heav
ing num ber to characterize heaving is proposed, and two p aram eterizations of
heaving for use in m ixed layer models are suggested.
1.2 T h e o c e a n ic su r fa c e b o u n d a r y la y er
1 .2 .1 O v e r v ie w
T h e ocean typically m anifests a surface boundary layer in which turbulent mixing driven from above by surface ffuxes of buoyancy and m o m en tu m and from below by shear a t the base of the layer creates a quasi-uniform stratification within
it. Thus it is conventionally referred to as a “m ixed layer” , although observations frequently reveal th e existence of velocity shear w ithin th e layer (e.g. Davis et
1. Introduction
al. (1981a), W eller and P lu eddem an n (1996)) indicating im perfect m ixing a n d /o r
the presence of organized stru c tu re s (e.g. Sm ith et al. (1987)).
T hroughout th e course of a year, the mixed layer executes a ch aracteristic seaaonal cycle, varying its thickness in response to th e forcing. In th e w inter, it
deepens because of h eat loss a t th e surface; in the spring progressively shallower
m ixed layers are form ed u n d er conditions of net ocean h eatin g u n til a new seasonal
therm ocline is form ed for th e year. O n top of this variability, each d iurnal period a shallow m ixed layer m ay be form ed in th e daytim e because o f stro n g p e n e tra tin g
solar heating, deepening at night back to its seasonal d ep th . T h e d e p th of m ixing can also vary w ith geographical location. Especially in th e w in ter, it could be as
little as 50 m etres in th e subtropics and as much as 500 m etres or m ore in th e northern N orth A tlan tic ocean.
This stu d y focuses on th e m id -latitu d e mixed layer, aw ay from eq u ato rial
and polar regions. M oreover, it considers open ocean conditions away from late ra l boundaries and horizontal variability, so th a t a one-dim ensional (1-D) m odel in
the vertical of local forcing and response can be adequately applied. In addition, salinity/freshw ater variability is not considered.
1 .2 .2 F o r c in g
The oceanic surface bo u n d ary layer is forced by fluxes o f h eat, m om entum ,
and freshwater.
The u p p er ocean gains h eat by absorption of solar rad ia tio n in th e daytim e,
and (norm ally) loses h e a t from th e surface due to tu rb u le n t tra n sfe r via sensible and latent heat h e a t fluxes, and due to long-wave radiation.
T h e sensible heat flux
Q h ~ P a P p a ^ ^ ’ (1-1)
where are air density a n d specific heat, respectively, results fro m a tu rb u
lent tran sfer of heat to /fro m th e atm osphere, depending on th e sign o f th e surface te m p e ra tu re difference betw een th e atm osphere and the ocean. Since th e air tu rb u
lence n ear th e surface is largely caused by shear, th e rate of sensible h e a t transfer
depends also on wind speed. T hus, on dim ensional grounds, is p aram eterized
by a “bu lk ” (i.e. involving averaged quantities) form ula (S m ith 1980; Large and Pond 1982; Sm ith 1988)
0 ^ = / > . c , . c „ i 7 „ ( r . - r j , (1.2)
w here is a dimensionless coefficient, the wind speed a t 10 m e tre s height, th e sea-surface tem p eratu re, th e 10-meter air tem p eratu re. U n d er m oderate
w ind conditions with wind speeds of 5 — 25 ms ~^ , which often prevail over the ocean, th e atm ospheric b o undary layer is nearly neutrally stratified . Different
values/ form ulae for the n eu tral have been suggested; e.g. S m ith et al. (1980)
suggests
lO^Cjj = <
0.83 for stab le conditions (T^ < Z)^),
(1.3)
1. Introduction.
Typical m ean values of Qf^ are in th e range 10 — 20 W m ~ ^ (N urser 1996), and could co n stitu te h e a t loss or heat gain by the ocean depending on th e sign of th e
air-sea te m p e ra tu re difference. T he la te n t h eat flux
Q^ = L y w q , (1.4)
where L y is th e la te n t heat of vaporization and q is hum idity, results from evapo
ratio n of w ater, ta k in g heat away from th e ocean. A gain on dim ensional grounds
th e evaporation r a te is param eterized as
E = P ^ c^U ^o {q^ -qJ , (1.5)
where Cg is a dim ensionless coefflcient, q^ is the satu ra tio n value of th e spe
cific hum idity a t te m p e ratu re T^, q^ is th e specific h u m id ity of air at 10 m etres.
S m ith (1980) suggests for th e n eu tral Cg
lO^Cg = 1.5 (1.6)
b u t later revises it to
(Smith. 1988). T he laten t heat of vaporization is
L y = 2.5008 X 10® - 2.3 x 10®T [Jkg (1.8)
where T is the w ater te m p e ra tu re (G ill 1982). Thus th e la te n t heat flux is
Q e ~ L y E = LyP^C^U^Q^q^ — q ^ ). (1-9)
A typical mean value of is 100 W m ^ or more (N urser 1996), and co n stitu tes heat loss by the ocean.
T he long-wave rad iatio n h eat flux is the result of em ission of infrared rad ia
tion by th e ocean. It is p aram eterized as a flux em itted by a black body, corrected for th e departure of th e ocean from a black body behaviour, for atm osphere mois
ture, an d for cloudiness. An exam ple is
= eo -r/(0 .3 9 - 0 .0 5 e / / ') ( l - 0 . 6 n /) , (1.10)
where e = 0.985 is th e e stim ated ocean emissivity, <j is S tefan ’s co n stan t, is th e
10-metre vapour pressure, is th e fraction of sky covered by clouds (G ill 1982). T ypical m ean values of in th e range 30 — 70 W m ~ ^ , and always represent
heat loss by the ocean.
Solar radiation is a source of heat for the ocean. T h e incom ing solar radia tion can be estim ated by em pirical form ulae involving th e solar co n stan t an d cor
1. In tro duction
Dobson an d S m ith 1988). F urtherm ore, th e insolation is absorbed w ith d e p th de
pen d in g on w ater clarity, w ith th e red sp ectral com ponents absorbed w ith in th e
u p p er 1 m etre of ocean, and the rest p e n e tra tin g deeper. P aram eterizations of th is ab so rp tio n profile involve an exponential d e p th dependence. This can be a single
ex ponential fit (Ivanoff 1977), th e m ore com m only used 2-exponent fit (P au lso n
an d Sim pson 1977), 3-exponent fit (W oods et a/.1984) or a fit w ith even m ore
ex p onents (Sim pson and Dickey 1981; W oods 1980). Using m ore exponents w ould typ ically im prove the accuracy of th e fit n ear th e surface, in the top 5 m e tre s of
ocean.
T h e ocean surface gains m o m en tu m by tu rb u len t transfer from th e a tm o sphere v ia th e w ind stress
r = —p j u u , (1.11)
v irtu a lly all of which goes into driving th e m ean current (R ichm an and G a rre tt 1977), im p a rtin g a “friction velocity”
“ ■
=O
w here is th e density of w ater. S ta rtin g w ith Taylor (1916), the drag of th e
where is a dim ensionless drag coefficient dependent on wind speed a n d on
atm ospheric stability. Different form ulae for the n eutral c^ have been suggested, e.g. S m ith (1980) suggests
10 c^ = 0.61 +0.063C/^jj for 6m s < < 22ms (1.14)
which is very close to a m ore recent form ula by Yell and and Taylor (1996). A typical w ind stress value is 0.1 N m ^ for of 8 — 9 m s
T h e u p p er ocean is also forced by a freshw ater flux, w ith rain an d ice m elting
being a freshw ater source, an d evaporation and freezing being a freshw ater sink for th e ocean.
1 .2 .3 P r o c e s s e s
T h e u pper ocean is host to a variety of processes, some of which are schem at
ically depicted in Fig. 1.1. Here I will describe briefly only th e processes considered in subsequent chapters.
C a b b e li n g
T h e equation of s ta te for seaw ater is nonlinear, th a t is to say th e density p is a nonlinear function of te m p e ra tu re T , salinity s, and pressure p (M illero
and Poisson 1981). For th e u p p er ocean a t tem p erate latitu d es th e te m p e ra tu re dependence is th e m ost im p o rta n t one. T h e tem p eratu re expansion coefficient
1. Introduction 10 breaking w av es Langm ulr circulation CONVECTION shear-induced turbulence breaking internal w aves
Fig. 1.1. Some of th e m echanism s affecting the tu rbulence in the oceanic sur face boundary layer: Langm uir circulation, breaking surface waves, shear-induced turbulence, convection, breaking internal waves (redraw n from T horpe (1985)).
nonlirieaxity m eans th a t if equal volumes of two w ater m asses of different proper ties are m ixed, the resulting m ixture, while having th e ir m ean tem p eratu re, will
have a d en sity larger th a n th e ir m ean density (Fig. 1.2). In th e special case when th e two w ater masses have different salinities and tem p eratu res b u t th e sam e or
nearly th e sam e density, th e resulting m ix tu re will be heavier th a n b o th and will
ten d to sink. A pparently W itte (1902) was th e first to recognize th e possible im portance of this phenom enon (Foster 1972), visualizing a frontal situ atio n when
th e two w ater masses are ju x tap o sed and la te ra l m ixing occurs, causing sinking in
a narrow region between th e m . T he sam e setup was considered also by G arrett and H om e (1978). However, cabbeling need not occur only in horizontal fronts.
Indeed, it occurs whenever th ere is m ixing of w ater of different properties, e.g. whenever th e m ixed layer entrains colder w ater from below. C abbeling m ight be
im p o rtan t for th e form ation of th e densest ocean w ater, e.g. in th e form ation of
th e A tlantic B o tto m W ater in th e Weddell Sea (Foster 1972). T h e salinity con trib u tio n to cabbeling opposes th a t of tem p eratu re, b u t is typically 2 orders of
m agnitude sm aller (K ihnatov and Kuzm in 1991).
L a n g m u ir c ir c u la tio n
In 1927, while crossing th e A tlantic Ocean, the chem ist Irw ing Langm uir observed floating seaweed arranged in streaks roughly parallel to th e wind di
rection, an d w ith a spacing of 100 to 200 m etres betw een th em . In subsequent
years he m ad e a series of experim ents in a lake, and showed (L angm uir 1938) th a t these streaks axe m anifestations of a subsurface organized stru c tu re — pairs of
I. Introduction 12 D1 I W c (D Q 02 T1 Tm T2 Temperature ■
Fig. 1.2. A schem atic of th e nonlinear dependence of density of seaw ater on tem p eratu re. W arm w ater expands m ore th a n cool w ater co n tracts. If w ater w ith te m p e ra tu re T l , d en sity D l , m ixes w ith w ater with te m p eratu re T 2 > T*l, density
D2 < D l , in equal volum es, th e m ixtu re, while of tem p eratu re T m = { T l + T 2 ) /2 ,
downwelling regions betw een the rolls (hence surface convergences an d streaks of flotsam above th em ). Langm uir concluded from his experim ents th a t these vor
tices, later n am ed after him , “ap p arently co n stitute the essential m echanism by
which th e [m ixed layer] is produced” . Furtherm ore, envisioning m ixed layer en- train m en t due to this Langm uir circulation (LC), he sta ted th a t th e “currents
thus set up a t th e b o tto m of th e [mixed layer] m ay sweep off th e u p p er p a rt of th e therm ocline, m ak in g it th in an d of increased gradient” .
A fter m a n y years of observations (in lakes and in the ocean, m ore recently in 1982 an d 1990 off th e coast of California (S m ith 1992; P lu ed d em an n et a/.1996)),
th e conceptual p ictu re for these subsurface vortices has rem ained largely as de scribed by Langm uir. Fig. 1.3 presents a schem atic diagram of LC by Pol
lard (1977). T h e spacing between streaklines, or “windrows” , can range from 2 to hundreds of m eters (Leibovich 1983). Observations (e.g. W eller an d P rice (1988)) and m odelling (e.g. Skyllingstad and Denbo (1995)) suggest th a t, as tim e pro
gresses, sm aller-scale cells are swept into th e larger scale, while a t th e sam e tim e
new sm all-scale cells form . T h e horizontal scale for the spacing, a t least of th e largest cells, is generally assum ed to correlate w ith the d ep th of th e therm ocline,
i.e. the cells have an aspect ratio of 0 (1) and penetrate to th e base of th e m ixed layer (ML). Som e observations (e.g. Sm ith (1992)) have confirm ed this, while o th
ers (e.g. W eller a n d Price (1988)) have found th a t LC is confined to th e u p p er half of th e ML. T h e localized downwelling velocities below the streaks are from 0.03 to
0.20 m s ^ (W eller and P rice 1988), sometim es m ore (Pollard an d T hom as 1989).
T h e upwelling is significantly slower an d broader. There is a c h aracteristic down wind surface horizontal je t. LC is always nearly aligned w ith th e w ind, an d if th e
1. Introduction 14
wind
c < 15w - 1-2 cm/s
-w - 10 cm/s
Fig. 1.3. A schem atic representation of Langm uir circulation (redraw n from Pol lard (1977)).
wind suddenly changes direction, LC gets reoriented in th e new direction w ithin a
few m inutes to a b o u t half an hour, w ith sm aller scale LC reorienting m o re quickly
th a n larger scale LC. These tim es are also indicative of the form ation tim es for LC (Leibovich 1983).
M any theories for the form ation of LC have been proposed, b u t th e com
m only accepted theory is th a t of C raik (1977) and Leibovich (1977a). It rehes on
an in teractio n between a sheared horizontal cu rren t and the surface waves, and involves an in stab ih ty of an infinitesim al dow nw ind je t w ith its induced vertical
v o rticity of altern atin g sign on b o th sides. T h e Stokes drift, which decays ver tically w ith d ep th , tilts these vorticities an d th u s creates the pairs of roUs w ith
downwelling in between. This p a tte rn gets reinforced, as w ater m oving tow ards th e surface convergence line gets accelerated by th e w ind stress which in tu r n leads
to m ore downwelhng, intensification of th e circulation, and stronger ho rizo n tal sur face je t.
I n t e r n a l w a v e s
G rav ity waves in the interior of th e ocean are com m onplace. In analogy to
surface grav ity waves, one can th in k of a d istu rb an ce of an interface betw een two layers of slightly different densities which will feel th e restoring force of gravity
(although m uch reduced because of th e sim ilar densities) and reverse direction,
overshoot its equilibrium position, and so on, resu ltin g in wave propagation. How ever, in tern a l waves can be supported in any continuously varying s ta b le s tra ti
fication. Because th e restoring force is m uch w eaker th a n in th e case of surface waves, in tern a l waves have spatial scales an d tim e scales which are ty p ically m uch
I. Introduction 16
larger. Away from the u p p e r ocean, typical internal waves have vertical displace m ents of th e order of tens of m eters and periods from tens of m in u tes to m any
hours (G a rre tt and M unk 1979). In the u p p er ocean, intern al waves typically have
sm aller am plitudes and sh o rte r periods (Pinkel 1975; Kase and C larke 1978; Dillon an d Caldwell 1980; Levine et a/.1983), b u t are still large com pared w ith surface
waves (Fig. 1.4). In tern al waves are com m only m anifested in single-sensor observa
tions as sh o rt period fluctuations in th e tem p e ra tu re (M unk 1981). N ansen (1902) m ight have been the first to rep o rt them . A ttrib u tin g these te m p e ra tu re fluctua
tions to in tern al waves, an d thus estim ating th eir instantaneous vertical velocity
by w = —{ d T f d t ) J { d T / d z ) assum es no significant horizontal advection a n d /o r diffusion (Briscoe 1975).
T h e equations for a Boussinesq fluid (Boussinesq 1903; Spiegel an d Veronis
1960) of varying stratificatio n w ith buoyancy frequency N [ z ) lead to th e dispersion relation (Phillips 1977)
ui = N cos 6 + f sin 0, (1.15)
where w is th e internal wave frequency, / is th e local inertial frequency, 6 is th e
angle to th e horizontal of th e vector w avenum ber. Varying 9 from 0 to tt/2 , thus
intern al waves can exist only in th e range of frequencies
Depth [m] Depth [m]
■drÆilLK
^ 9 ^
Fig. 1.4. Ten-day tim e series of isotherm s versus d ep th and tim e, tak en from the LOTUS d a tase t (Briscoe and W eller 1984; Bowers et al. 1986). T he high-firequency variability in th e isotherm s’ position in an d below th e seasonal therm ocline at 20
I. Introduction 18
G arrett and Munk (1975), using m any different observations o f internal wave activ ity in th e ocean away from th e upper layer, compiled a representation
of th e d istribution of internal wave energy in wavenumber and frequency space, subsequently called the GM spectrum . A fter tests and refinements it was con cluded th a t the observed characteristics of deep-ocean internal waves a re nearly
universal in space and tim e, thus the GM spectrum can be regarded as a universal equilibrium spectrum , the result of a dynam ic balance between wave generation,
in teractio n , and dissipation. In th e u p p er ocean, where such an equilibrium is
not to be expected due to th e forcing an d variability, the internal wave field of higher frequencies deviates from th e GM spectrum , and exhibits high coherence
w ith d ep th (Fig. 1.4).
Internal waves can be generated by a variety of mechanisms. Surface gen e ratio n by the atm osphere could involve travelling pressure fields/ fronts (Keller
an d M unk 1970; Polyanskaya 1969; Leonov and Miropolsky 1973), a travelling
buoyancy flux (M agaard 1973), and a travelling wind stress field (Tom czak 1966;
Tom czak 1967; Krauss 1972a; Krauss 1972b), the la tte r being th e m ost plausi ble source (Thorpe 1975). Surface generation by interaction of pairs of surface waves w ith sim ilar wavenumbers and frequencies is also possible (T h o rp e 1966;
N esterov 1972; Brekhovskikh et a/.1972). T h e steady flow of th e stratified ocean
over b o tto m topography is another source (H uppert and Miles 1969; B ell 1975),
generating standing internal waves analogous to lee waves in the atm osphere. T idal (barotropic) currents over topography can also generate internal waves of tid a l fre quency (Cox and Sandstrom 1962; Baines 1973; BeU 1975; Baines 1982). In the
ocean interior, in tern a l waves can be g en erated a n d /o r modified by wave-wave in teractions (O lbers 1976; McComas and B re th e rto n 1977; Pom phrey et a/.19S0).
D issipation of in tern al waves could occur th ro u g h sh ear in stab ih ty (T horpe 1973)
an d wave b reaking (O rlanski and Bryan 1969; O rlanski and Ross 1973; G argett and Holloway 1984), and also through viscous a tte n u a tio n (LeBlond 1966).
1 .3 O b se r v a tio n s
I will briefly describe th e sets of observations of th e upper ocean which are
referred to in subsequent chapters.
1 .3 .1 O W S P a p a
Since 1952 (T ab ata 1965), extensive oceanographic observations of the up
p e r ocean were m ade by Canadian w eatherships a t O cean W eather S tatio n (OW S) “P ” (50°N, 145°W ) in th e northeast Pacific O cean, u n til the w eatherships were
retired in 1981. T h e d a ta consist of m eteorological observations and b a th y th e r
m ograph (B T ) te m p e ra tu re records over m any years a t 3-hour intervals (th e BT records are a t 3-hour intervals since 1965). T h e m eteorological observations consist
of w ind speed, w ind direction, sea-surface te m p e ra tu re , air tem p eratu re, w et-bulb
a n d dew -point tem p eratu res, cloud cover, an d surface air pressure. T h e B T tem p e ra tu re d a ta are generally a t 5-metre intervals from th e surface to 295 m.
I. Introduction 20
1 .3 .2 L O T U S
T h e Long-Term Upper-Ocean S tudy (LOTUS) experim ent was conducted from M ay 1982 to M ay 1984 a t 34°N, 70°W in the deep w ater over th e H at
te r as abyssal p lan e in the Sargasso Sea in th e western su b tro p ical A tlantic Ocean.
T h e d a ta consist of m eteorological observations, tem p e ra tu re and current m eter records (Briscoe an d Weller 1984; S tram m a et a/. 1986). T hese are divided into
four periods of several m onths each, w ith gaps in between w hen th e moorings were
replaced an d w ith o th e r gaps caused by d a ta losses. T h e m eteorological observa
tions consist of w ind speed, wind direction, sea-surface te m p e ra tu re (at a d ep th of
0.6 m ), air te m p e ra tu re , barom etric air pressure, and directly m easured insolation. T he hum idity m easurem ents were unreliable. Mooring d a ta consist of te m p e ra tu re
and east an d n o rth currents generally a t depths of 5, 10, 15, 20, 25, 35, 50, 65, 75,
100 m etres and som e in deeper w ater. D ata are available a t 15-m inute intervals.
1 .3 .3 M IL E
T h e M ixed Layer E xperim ent (M ILE) was conducted from A ugust 19 to
Septem ber 6, 1977, a t 49 37 N, 145°6^W, in the vicinity of OWS Papa. T he d a ta
consist of m eteorological observations, tem p eratu re and cu rren t m eter records (Davis
et al.l981a), a n d also two series of casts through the u p p er 200 m etres of ocean
w ith a m icro stru c tu re profiler (Dillon and Caldwell 1980). T h e meteorological observations consist of wind speed, wind direction, sea-surface te m p eratu re, air
tem p eratu re, w et-bulb and dew-point tem peratures, surface air pressure, cloud cover, m ade every 3 hours, and solar radiation m easurem ents m ade every 1 hour.
T h e MILE-1 m ooring has com plete 19-day records of tem p eratu re and east an d n o rth currents a t depths of 11, 23, 26, 29, 32, 35, 41, 44, 47, 50, 75, 92, 125 an d
175 m etres and 16-day records at 20 an d 38 m etres. T he MILE-2 mooring has
com plete records of tem perature and east and n o rth currents a t depths of 5, 10, 15, 20, 42 and 54 m etres. T he d ata are available a t 112.5-second intervals.
1 .3 .4 P A T C H E X
T he PATCHes E x p e rim en t (PA TCH EX ) was conducted in October 1986
aro u n d 34°N, 127°W , 800 km west of P oint Conception, off th e coast of California.
T h e d a ta include meteorological observations, te m p e ra tu re and horizontal velocity
records (B rainerd and Gregg 1993). T h e meteorological observations consist of wind speed, w ind direction, sea-surface tem p eratu re, air tem p eratu re, wet-bulb
and dew -point tem peratures, cloud cover, surface air pressure, and insolation. An A dvanced M icrostructure Profiler recorded 11 days of tem p eratu re, conductivity,
te m p e ra tu re gradient, and velocity fluctuations profiles down to 300 m, at an
average ra te of 2.4 profiles per hour. Relative current velocities were m easured using an acoustic current m eter, a t a ra te of 4 drops per day for th e last 7 days.
1 .3 .5 G A T E , p h a se I II
T h e G A R P A tlantic Tropical E xperim ent (G A TE), phase III, was con
d u cted from A ugust 30 to Septem ber 19, 1974, a t 9°N, 23°W in the A tlantic O cean, to stu d y in ternal waves in th e u p p er ocean (Kase an d Clarke 1978). T h e
d a ta consist of a triangular horizontal array of moorings, recording tem p eratu re and currents, an d GTD profiles from a drifting ship. T h e CTD profiles reached
1. Intro ductio n 22
down to 500 m , at 3-hour intervals, and th e re were also som e sh o rt-term CTD
series from 20 to 70 m, a t 108-second intervals. The d a ta a t th e m oorings are at 225-second intervals.
C h a p t e r 2
M ix ed layer m o d e ls
I will present the approxim ate and sim plified governing equations th a t are ad eq u ate for use in 1-D m odels of th e u p p er ocean, and describe in m ore detail
m odels referred to in subsequent chapters.
2 .1 E q u a ti o n s
T h e full equations governing th e evolution of the ocean are (G ill 1982):
T h e mass conservation equation
P ^ + V - u = 0, (2.1)
w here D / D t = + u • V is the m ateria l derivative, u is th e fluid’s (vector) velocity, an d p th e density of th e fluid.
T h e m om entum equation
^ + / ( k X u) = p V p - g + f/V^u, (2.2)
where / is th e Coriolis p a ra m e te r/in ertia l frequency, k the vertical co ordinate u nit
vector, p th e pressure, g th e gravitational acceleration, u the kinem atic viscosity
2. M ixed layer m odels 24
T h e heat equation
~ W ^ + S h . (2-3)
where T is th e te m p eratu re, th e specific h eat a t constant pressure, a = —I j p l d p j d T ) th e tem p eratu re expansion coefficient a t constant safinity and pressure, % the th er
m al conductivity, F th e rad iativ e flux, th e heating due to a change of phaae, chemical reaction, and viscous dissipation.
T he salinity equation
Ds
p — = V • (Kj^Vs), (2.4)
where s is th e salinity, an d is th e salinity diffusivity.
The equation of s ta te
p = p { p , s , T ) . (2.5)
We can sim plify these equations by m aking a num ber of approxim ations
a n d /o r assum ptions based on th e context in which they will be used, th a t is, for
upper ocean modelling. Because b o th particle and phase speeds of disturbances are much sm aller th a n th e speed of sound, an d also because th e vertical scale
of the m otion is sm all com pared to the scale height of th e ocean, we assume incom pressibility (B atchelor 1967), so D p / D t = 0 and therefore (2.1) reduces to
th e continuity equation V • u = 0; we also ignore pressure variations in the heat equation, and assum e = const. Because th e reference h y d ro statically balanced
density varies little around its m ean, and also because the m otion-induced density variations are m uch sm aller th an the reference state density, we can m ake the
Boussinesq approxim ation (Boussinesq 1903: Spiegel and Veronis 1960) and replace
density in th e equations w ith a constant reference density except when it gives
rise to buoyancy (6 = —gp^ ^(p~~Pq)} forces in the gravitational acceleration term .
In addition, we make th e boundary layer approxim ation th a t vertical gradients are
m uch g reater th an horizontal gradients, and in fact com pletely drop horizontal
gradients, assuming a local balance in the vertical as 1-D ML m odels do (Niiler an d K raus 1977). We are interested in the evolution of m ean properties, and how th ey are affected in a m ean sense by turbulence, so we perform Reynolds
decom position of all variables into mean and fluctuation parts. M olecular term s
are also neglected as sm all com pared with th e other term s. Finally, th e averaged equations for m ean qu an tities are
W = _ / ( k X Û ) - | ( ; 7 7 ) (2.8)
P = p{s,T ), (2.9)
where overlines denote averaging and mean quantities, and prim es denote turb u len t fluctuations. I[z) is th e vertical absorption profile for th e insolation. In addition
to these equations, th e equation for th e averaged turbulent kinetic energy (TK E) is also often used for calculating th e am ount of mixing a n d /o r th e entrainm ent
2. M ix ed layer m odels 26
ra te for a m ixed layer. It is, u n d er th e sam e assum ptions and approxim ations as above (K raus 1972),
1 d(q^) ' ' d U ~!~r d >, 2 , T , .
2 ~ d T = + ^ 6 - - ^ w {q ( 2 + p / p j - e, (2.10)
where = u ^ + to ^ is tw ice th e T K E per unit mass. T h e first te rm on th e
rig h t-h an d side of (2.10) is th e ra te of work by the Reynolds stress w u o n th e m ean shearing flow j d z \ th e second te rm is the rate of work of buoyancy forces;
th e th ird term is the convergence of th e tu rbulent vertical flux; the fo u rth te rm is th e dissipation rate of T K E . H enceforth overlines denoting m ean q u an tities will be dropped, but implied.
2 ,2 T h e P r ic e - W e lle r - P in k e l m o d e l
T h e model of Price, W eller, an d Pinkel (1986), henceforth th e P W P m odel,
is a 1-D bulk m ixed layer m odel, i.e. it assumes the existence of a w ell-m ixed surface layer. T he m odel solves equations (2.6 — 2.9) on a fine uniform v ertical
grid (e.g. w ith A z = 1 m ), and employs stab ility criteria to establish, m ain tain ,
and evolve a mixed layer of uniform tem p eratu re, salinity, m om entum , an d density. In accord w ith the scope of this study, let us consider the salinity to be co n sta n t. T h e surface boundary conditions axe
w 'r'(o ) = 0L (2.11)
1(0) = (2.12)
IÜ u (0) = r (2.13)
w here = Q[, + + Q is th e n et heat flux (loss) from long-wave radiation, la te n t, eind sensible fluxes, respectively; Q^ is th e incoming solar radiation; t is th e
vector wind stress. T h e tu rb u le n t fluxes at d epth are not e x p h citly calculated, b u t
are im plicitly accounted for as a result of th e adjustm ents by s ta b ility criteria. T h e
absorption profile I{ z) for th e p en etratio n of solar radiation a t d ep th is assum ed to have a double exponent dependence (Paulson and Sim pson 1977), i.e.
I { z ) = 1(0) [ r e x p ( z / ^ J - b ( l - r) exp(z/^^)] , (2.14)
so th e rad iatio n is divided into a long-wave and a short-wave sp ectral com ponent w ith a tte n u a tio n depths of and respectively. The a tte n u a tio n depths change
w ith w ater tu rb id ity . Jerlov (1976) designated several w ater ty p es based on w ater
chlorophyll concentrations. For a fairly clear, mid-ocean w ater of ty p e lA th e values are r = 0.62; = 0.6m; = 20m .
For each tim e-step , th e P W P m odel proceeds as follows.
(i) Insolation is absorbed in the water colum n, an d net heat loss
occurs from th e top grid-point, or layer, thus changing th e te m p e ra tu re profile. D ensity is calculated (from a linear or nonlinear equation of s ta te , depending on
im plem entation).
(ii) A sta tic stab ility criterion
2. M ix e d layer m odels 28
is evaluated, an d com plete m ixing of tem p eratu re, m om entum , and density occurs from th e surface downwards, iteratively, until a sta tic a lly stable stratification w ith a m ixed layer on top is achieved. Mixing at this stage is equivalent to th a t caused
by free, n o n -penetrative convection.
(iii) T h e wind stress is applied to th e to p grid-point, and m om entum
is d istrib u te d uniform ly through the m ixed layer. H orizontal currents are ro ta te d inertially.
(iv) Overall m ixed layer shear in sta b ih ty versus buoyancy sta b ih ty
is ev alu ated a t th e base of th e mixed layer by a bulk Richardson num ber criterion
Rk = > 0.65 (2.16)
where h is th e m ixed layer d ep th , 0.65 is the critical Rb value, A p and | A u | are th e
density an d velocity differences, respectively, betw een the ML and ju st below it. If Rb < 0.65, th e m ixed layer deepens by one grid-point and properties are m ixed
com pletely w ithin th e new ML. The new Rb n u m b er is then evaluated, and en
tra in m e n t and m ixing proceed iteratively until th e criterion is satisfied. This ty p e
of d y nam ic sta b ility criterion, implying th a t the ML is always either stable or in a sta te of m arg in al stabihty, has also been im plem ented by others (Pollard et a/.1973; Price et aL1978; K ronenburg 1985). Although no sound theoretical evidence th a t a
tru e critica l R ichardson num ber exists for a tu rb u le n t m ixed layer (Philhps 1977),
lab o rato ry experim ents by EUison and Turner (1959) show th a t entrainm ent by a surface half-jet decreases by a fuU order of m ag n itu d e as Rb increases from 0.4 to 0.8, suggesting a n approxim ate critical bulk R ichardson num ber in th a t range.
(v) Local shear flow stability is evaluated below th e m ixed layer w ith a gradient Richardson num ber criterion
where 0.25 is th e critical Rg value; gradients are evaluated by taking first differences
over neighbouring grid-points. If R g is found to be less th an 0.25 a t a given pair
of grid-points, p a rtia l m ixing of properties occurs betw een them . If densities prior to the partial m ixing are and the density exchanged is e.g.
(2-18)
w ith a com plete m ixing for Rg = 0; a constant (0.3) slightly larger th a n 0.25 is used to hasten convergence, the larger value having no appreciable effect in the
solutions (Price et a/.19S6). Then the Rg array is recalculated, reevaluated, ajid
new adjustm ents are m ade, until Rg > 0.25 everywhere.
The m o tivation for adding gradient Richardson num ber dependent m ixing
below th e m ixed layer was to sm ooth out sharp te m p era tu re ju m p s at th e ML base, m aking m odel profiles more like the profiles from observations, and also to
relieve local shear instabilities in th e stratified fluid below the ML. However, the im plem entation of this criterion in the model leads to “unm ixing” of th e m ixed layer established w ith the bulk Richardson num ber criterion, as p artial m ixing
adjusts the te m p e ra tu re at the bo tto m ML grid-point as well. As a result, the final ML dep th diagnosed by the m odel is not the ML dep th established by overall
2. M ix ed layer m odels 30
m ixed layer stability, as evident e.g. from Fig. 2 of Schudlicb and P ric e (1992) and th e accom panying figure caption. Thus, the Rg procedure seems to deliver
m ore th a n it was designed to do; this does not seem to be com m only recognized, as th e m odel has been described or im plied by others as setting its ML d e p th w ith
a bulk criterion, w ith R g m ixing only resolving instabilities below such a m ixed
layer (A rcher et a/.1993; K a n th a an d Clayson 1994; Large et a/.1994; L arge an d Craw ford 1995).
M odel diagnostics (not shown) revealed th a t to satisfy R g > 0.25 every
where, up to tens of thousands of ad ju stm en ts at each tim e-step m ight b e needed for a region w ith a few tens of grid-points. In addition, while looking o n occasion
a t spikes in th e P W P m odel o u tp u t such as ML dep th and sea-surface te m p e r
atu re, I was able to trace th e m back to th e Rg criterion procedure. It appears th a t, afte r m any sm all ad ju stm en ts, th e final ML d ep th m ight in co rp o rate a m ore
or less a rb itra ry fluctuation. A pparently, th e problem of m axim izing Rg , by over
shooting discrete-step ad ju stm en ts, to achieve Rg > 0.25 everywhere for a n arra y of grid-points, does not have a unique solution. So th e final ML d e p th m ay not
be consistent from one tim e-step to th e next as different “solutions” are realized. T he spikes disappear, however, if th e ML dep th achieved from overall m ix ed layer
sta b ih ty w ith th e bulk Richardson num ber criterion is enforced, while s till allow
ing for ad ju stm en ts in th e therm ocline below it^. Therefore, th e m ixed lay er d e p th
^This can be done as follows. For each iterative adjustm ent due to the gradient R ichardson num ber criterion, i f partial m ixing aifected the b o tto m grid-point o f the m ixed layer established b y th e R h
criterion, a com p lete m ixing is performed w ithin this layer to restore its hom ogeneity before th e n ext iteration is a ttem p ted .
based on overall stab ility is consistent in tim e, and in fact it seems difficult to ju stify m odifications of th is d ep th via th e gradient Richardson nu m b er criterion.
2 .3 T h e M e llo r - Y a m a d a L e v e l 2 m o d e l
A hierarchy of tu rb u le n t closure m odels, based on an order-of-m agnitude
analysis of sm all deviations of th e Reynolds stresses and heat fluxes from local isotropy, was described by M ellor and Y am ada (1974), refined by th em la te r (MeUor
a n d Y am ada 1982), and reexam ined by G alperin et al. (1988). In the lim iting case of neax isotropy, algebraic relations axe o b tain ed for all tu rb u le n t q u an tities. The
resulting m odel, classified as Level 2 closure (MY-2), is a ttra c tiv e because of its
sim plicity and robustness (G alperin et a/. 1988), and has been applied in various situ atio n s (M artin 1985; M a rtin 1986; K an th a an d Clayson 1994; Large et a/.1994). T h e equations, in term s of buoyancy instead of te m p e ra tu re and salinity separately,
are d U d d t = - / ( k X U ) - — (2.20) - r - T d V — V — = —w u • -5---- \-w b (2.21) A.. d z
w here (2.21) is the averaged T K E equation (2.10), simplified by neglecting the
convergence of the tu rb u le n t vertical flux, and assum ing a balance betw een shear p ro d u ctio n , buoyancy forcing, and dissipation. T h e dissipation is assum ed to be
2. M ix e d layer m odels 32
“m a ste r length scale” of turbulence, to which all o th e r relevant length scales are
p ro portional. T his m aster scale is calculated from B lackadar’s (1962) interpolation fo rm u la
w here z is distance from a boundary, k % 0.4 is von K arm an ’s constant, is a
m easu re of th e extent of th e turbulent field. T h e form ula interpolates between
I — > H.\z\ as |z|/Zg — V 0, and I — y as — j- oo; is calculated from
w here q is rm s tu rb u len t velocity, and 0.2 is the chosen value of th e em pirical
co n sta n t in agreem ent w ith results from higher level MY models (Mofjeld and
Lavelle 1984; M artin 1985).
T h e tu rb u len t fluxes in (2.19 — 2.21) are param eterized by eddy diffusivity a n d eddy viscosity
w b = - K ^ { z ) ^ = - I q S ^ ^ (2.24)
w here th e vertically-varying eddy coefficients are p roducts of the m aster length
scale Z(z), rm s tu rb u len t velocity g(z), and a gradient Richardson num ber Rg de p e n d e n t sta b ility functions involving more p roportionality constants. A fter
su b stitu tin g the values of th e proportionality constants tak en from Mellor and Ya-
m ad a (1982), wherever needed, th e final equations are
8
-
i
(«.
8
) -
0.
5 _ ________0.4 0- 3.0 8 0^ _________
M (1 - 34.680H)(1 - 6.130^) ^
where = —{l/q)^db/dz, and I and are calculated from (2.22) and (2.23) above.
It can be shown th a t 6"^ > 0, > 0, and th a t they are zero for Rg > 0.19.
2 .4 O th er m o d e ls
One-dimensional m odels of the m ixed layer generally fall into two classes. Turbulence closure ML m odels (also called K-theory m odels) can be organized
and classified as parts of a single hierarchy (Mellor and Y am ad a 1974; Mellor and
Y am ada 1982; Galperin et a/.1988) based on an order of m ag n itu d e analysis of sm all deviations of the second m om ents of turbulence from th e s ta te of local isotropy.
T hese models employ vertically-varying eddy diffusivity an d ed d y viscosity. T h e
M ellor-Yamada Level 2 m odel described in the previous section is an exam ple of a K -theory model. Details on other K -theory models will n o t be given here; th e
reader is referred to th e th ree papers cited above. Bulk ML m odels assume th e
