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References 1) (HIG Notes) McCreary, J.P., 1980: Modeling wind-driven ocean circulation. JIMAR 80-0029, HIG 80-3, Univ. of Hawaii, Honolulu, 64 pp. 2) EquatorialNotes.pdf 3) Shankar_notes/equatorial_ocean.pdf 4) McCreary, J.P., 1981b: A linear stratified ocean model of the Equatorial Undercurrent. Phil. Trans. Roy. Soc. Lond., 298A, 603– 635. 5) McCreary, J.P., 1985: Modeling equatorial ocean circulation. Ann. Rev. Fluid Mech., 17, 359–409.
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Wind-forced solutions:
Equatorial ocean A short course on: Modeling IO processes and
phenomena INCOIS Hyderabad, India November 1627, 2015 References 1)
(HIG Notes) McCreary, J.P., 1980: Modeling wind-driven
oceancirculation. JIMAR , HIG 80-3, Univ. of Hawaii, Honolulu,64
pp. 2) EquatorialNotes.pdf 3) Shankar_notes/equatorial_ocean.pdf 4)
McCreary, J.P., 1981b: A linear stratified ocean model of
theEquatorial Undercurrent. Phil. Trans. Roy. Soc. Lond., 298A, 603
635. 5) McCreary, J.P., 1985: Modeling equatorial ocean
circulation. Ann.Rev. Fluid Mech., 17, 359409. Equatorial
phenomena: quasi-steady currents
An Atlantic section along 23W (https://www.sfb754.de/l-atalante). A
deeper Pacific section along 159W, showing deep equatorial jets
(Ascani et al., 2010; JPO). Large-scale coastal phenomena forced by
steady y include: i) alongshore coastal currents, namely, a surface
jet in the direction of the wind and an opposite-flowing CUC; and
onshore/offshore flow, which for upwelling-favorable winds consists
of onshore flow at depth, coastal upwelling, offshore Ekman drift.
Coastal phenomena forced by variable y are the radiation of
coastally trapped waves along the coast and of Rossby waves
offshore. Shelves also impact coastal phenomena, for example,
allowing shelf waves and inhibiting the offshore propagation of
Rossby waves. Coastal flows are also driven by Q.For example, when
Q cools T(xe,y,0) poleward, Kelvin-wave adjustments cause the
surface layer to thicken poleward roughly quadratically (Sumata and
Kubokawa, 2001). There is a remarkable set of near-equatorial
quasi-steady currents in the Pacific Ocean. They include the
surface westward SEC, and the subsurface, eastward, EUC and
Tsuchiya Jets (SEUC & NEUC). Equatorial phenomena: quasi-steady
currents
23W An Atlantic section along 23W
(https://www.sfb754.de/l-atalante). A deeper Pacific section along
159W, showing deep equatorial jets (Ascani et al., 2010; JPO).
Large-scale coastal phenomena forced by steady y include: i)
alongshore coastal currents, namely, a surface jet in the direction
of the wind and an opposite-flowing CUC; and onshore/offshore flow,
which for upwelling-favorable winds consists of onshore flow at
depth, coastal upwelling, offshore Ekman drift. Coastal phenomena
forced by variable y are the radiation of coastally trapped waves
along the coast and of Rossby waves offshore. Shelves also impact
coastal phenomena, for example, allowing shelf waves and inhibiting
the offshore propagation of Rossby waves. Coastal flows are also
driven by Q.For example, when Q cools T(xe,y,0) poleward,
Kelvin-wave adjustments cause the surface layer to thicken poleward
roughly quadratically (Sumata and Kubokawa, 2001). and a similar
set exists in the Atlantic Ocean. Equatorial phenomena:
quasi-steady currents
23W An Atlantic section along 23W
(https://www.sfb754.de/l-atalante). A deeper Pacific section along
159W, showing deep equatorial jets (Ascani et al., 2010; JPO). 159W
Further, there is a remarkable set of deeper currents as well.
Nearly steady currents like those in the Pacific and Atlantic dont
exist in the Indian Ocean because the monsoon winds are highly
variable. Equatorial phenomena: El Nino
Sea level movie A similar ocean transition occurs during IOD
events, with an IOD analogous to La Nina. La Nina in the Pacific is
analogous to a positive IOD event in the Indian Ocean. Equatorial
phenomena: Indian Ocean
Sea level movie In the Indian Ocean, steady equatorial currents are
weak because there is almost no steady component to the wind
forcing. As a result, the most prominent feature of the equatorial
currents are the semiannual eastward flows (Wyrtki Jets). Questions
What forcing mechanisms drive equatorial currents?
zonal and meridional wind stress What are equatorial waves?
equatorial gravity, Rossby, and Kelvin waves; mixed Rossby/gravity
(Yanai) wave How do they differ from midlatitude waves? dynamically
very similar; extra Yanai wave; discreteness What are the key
differences between 2-d and 3-d theoriesof equatorial circulation?
Yoshida Jet; establishment of px to balance x How do equatorial
waves reflect from basin boundaries? Kelvin- and Rossby-wave
reflections; critical latitude Some questions and possible answers.
Introduction Equatorial waves Solutions for switched-on winds
Solutions for periodic winds Equatorial waves Equatorial-ocean
equations
Equations for the un, vn, and pn for a single baroclinic mode are
(1) Because f vanishes at the equator, no terms can be dropped that
allow for mathematically simple solutions near the equator. A
useful assumption, though, is to set f = y, known as the equatorial
-plane approximation. As a result, one can look for solutions as
expansions in Hermite functions. Equatorial gravity and Rossby
waves
We look for free-wave solutions to (1) of the form, (y)exp(ikx it),
without damping (A = 0), and, for convenience, we drop the
subscript n.The resulting v equation is (2) (2) It is convenient to
introduce the non-dimensional variable and to rewrite the v
equation in terms of . The mathematical difficulty with obtaining a
dispersion relation from (2) is that, because f varies so much near
the equator, it is not possible to set () = exp(iy).Rather, (y) is
the set of solutions (eigenfunctions) that satisfy where = 0, 1, 2,
.They are referred to as Hermite functions. Reinserting subscript
n, the length scale Rn = (on)1 = (/cn) is referred to as the
equatorial Rossby radius of deformation.Note that it has a
different value for each baroclinic mode n.Usually, its reported
value is for the n = 1mode. With cn = 250 cm/s and = 2.28x1013
cm1s1, its value is R1 = 331 km. NOTE: I use in two different
waves.In previous talks, it was the meridional wavenumber.Here, it
is the index of a Hermite function and (as we shall see) an
equatorial Rossby or gravity wave. Equatorial gravity and Rossby
waves
The figure plots the first six Hermite functions ( = 05).The
scaling factor, LR = Rn = (cn/), the equatorial Rossby radius of
deformation for baroclinic mode n.(For n = 1, LRis roughly 331
km.)Note that the are less equatorially trapped (extend farther off
the equator) as increases.Note also that they alternate between
being symmetric and antisymmetric about the equator. For large ,
the Hermite functions resemble cosine or sine curves near the
equator.They begin to decay at latitudes higher than the turning
latitude.So, the Hermite functions are equatorially trapped. Ascani
(2002) Fedorov and Brown (2007) High-order Hermite functions are
similar to sine waves, except that they cut off beyond a certain
latitude, the turning latitude. They cut off when 2 becomes bigger
than 2 +1.In that case the curvature of the response changes sign
and the response becomes exponential rather than oscillatory.
Equatorial gravity and Rossby waves
The solutions to (2) can be represented as expansions in Hermite
functions (3) where v is a wave amplitude.Each term in expansion
(3) is an individual equatorial wave. Inserting term in (3) into
(2) gives (2) which provides the dispersion relation for
equatorial, Rossby and gravity waves. Equatorial gravity and Rossby
waves
The dispersion relations for equatorial and midlatitude waves are
very similar.They differ only in that = f/c varies continuously for
midlatitude waves, whereas has discrete values for equatorial
waves. For each > 1, there is a gravity wave (large ) and a
Rossby wave (small ).The plot shows waves for = 1, 2, and 3. /o k/o
1 3 For = 0, there is a new type of wave, the mixed Rossby-gravity
(Yanai) wave, which joins the Rossby-wave (gravity-wave) wave
curves for large negative (positive)values of k. Matsuno first
published and discussed this famous dispersion relation. Mixed
Rossby-gravity (Yanai) wave
The curious form of the Yanai-wave dispersion curve happens because
it factors into two parts when = 0.We have Mixed Rossby-gravity
(Yanai) wave
The curious form of the Yanai-wave dispersion curve happens because
it factors into two parts when = 0.We have The second factor
describes a wave that travels westward at the speed of a Kelvin
wave.It can be shown that this wave blows up at , and so it must be
discarded. The single dispersion relation for the Yanai wave is
then For small and large values of , the relation simplifies to,
the same properties for Rossby and gravity waves, respectively.
Equatorial gravity and Rossby waves
The v, u, and p fields for equatorially trapped Rossby and gravity
waves are whereV is a constant amplitude, and j = 1 (2) corresponds
to the (+) sign. Equatorial gravity and Rossby waves
k/o 1 3 The u, v, and p fields for a Yanai wave when cn = 250 cm/s
and P = 30 days.For this P, /o = .36 and = 7.3. Courtesy of
Francois Ascani Equatorial gravity and Rossby waves
k/o 1 3 The u, v, and p fields for a Yanai wave when cn = 250 cm/s
and P = 360 days.For this P, /o = .03 and = 0.64. Courtesy of
Francois Ascani Equatorial gravity and Rossby waves
k/o 1 3 The u, v & p fields for an = 1 Rossby wave when cn =
250 cm/s and P = 360 days.For this P, /o = .03 and = 240. Courtesy
of Francois Ascani Equatorial gravity and Rossby waves
k/o 1 3 The u, v, and p fields for an = 2 Rossby wave when cn = 250
cm/s and P = 360 days.For this P, /o = .03 and = 140. Courtesy of
Francois Ascani Equatorial Kelvin wave
The equatorial Kelvin wave has v = 0, and so was missed in the
preceding solutions.To find it, set v = A = 0 in (1), and look for
a free-wave solution of the form (y)exp(ikx it). With these
restrictions, equations (1) reduce to The first and third equations
imply and the second then gives (4) Equatorial Kelvin wave
The solution to (4) is The solution that grows exponentially in y,
which corresponds to the root, k = /c, is physically unrealistic in
an unbounded basin and must be discarded.Therefore, the only
possible wave is (5) The y-structure of the Kelvin wave is o(y),
the lowest-order Hermite function. which describes the structure
and dispersion relation for the equatorial Kelvin wave.In (5), I
have used the property that and redefined the arbitrary constant
amplitude to be Po = P'o. Theoretical equatorial waves
To summarize, for each > 0, there is a gravity wave (large ) and
a Rossby wave (small ).The plot indicates the waves only for = 1,
2, and 3. /o k/o 1 3 There is a mixed-Rossby-gravity wave for = 0,
that behaves like a Rossby (gravity) wave for k positive
(negative). There is an equatorial Kelvin wave. A lot of
mathematics led to this set of dispersion curves.Do any of these
waves actually exist?! Observed equatorial waves
Tom Farrar (2010) The first equatorially trapped waves to be
discovered were gravity-wave resonances with periods of O(10 days)
(Wunsch and Gill, 1976; Deep-Sea Res.).There are no publications
that explore the possibility of Rossby-wave resonances. /o k/o 1 3
The equatorial Kelvin wave was discovered after it was predicted
(Knox and Halpern, 1982, JMR). The mixed Rossby-gravity (Yanai)
wave was first observed in the atmosphere by Yanai.In the ocean, it
was (probably) first detected in the Indian Ocean by Reverdin and
Luyten (1986) using altimeter data. Movies E Who first detected an
equatorial Rossby wave? Solutions for switched-on winds
x-independent (2-d) Yoshida Jet
Kozo Yoshida wrote down the first solution for an x-independent
(2-d) equatorial current driven by zonal winds.The (more complete)
theoretical solution developed somewhat later (Dennis Moore) has
come to be called the Yoshida Jet (Jim OBrien). The basic dynamics
of the Yoshida Jet can be understood from the zonal-momentum
equation. Neglecting the pressure-gradient and mixing terms in the
zonal momentum equation gives Offshore, Ekman balance (fvn = x/Hn)
holds, whereas at the equator un continues to accelerate (unt =
x/Hn).The switch from one dynamical regime to the other occurs at y
on = (/cn) . Bounded (3-d) Yoshida Jet
Zonal flows along the equator (Yoshida Jets) in reality and models
dont continue to accelerate.Why not? Because in the real world
either the wind forcing or the ocean basin is zonally bounded,
which introduces x-dependence into the solution.(An exception is
the Southern Ocean, but we will not consider that case here.) For
convenience, we can still drop the mixing terms in the zonal
momentum equation, and at the equator the Coriolis term
vanishes.The boundaries, however, introduce x-dependence so we
cannot neglect the pnx term In this case, the system can stop
accelerating by adjusting to a state where the pressure gradient
balances the wind. It does so by radiating equatorial Kelvin and
Rossby waves. Bounded (3-d) Yoshida Jet
d (1 month) d (6 months) Equatorial jet Kelvin wave Rossby wave
Coastal KW Rossby wave Kelvin wave 1) These process occurs
throughout the year in the Indian Ocean, in response to the
semiannual oscillation of the equatorial winds. What happens when
the basin boundaries are included? Suppose that the ocean basin is
unbounded but the wind is bounded, a patch of zonal wind. In
response to forcing by a patch of easterly wind, an
acceleratingYoshida Jet initially develops in the forcing
region.Subsequently, KWs and RWs radiate from the forcing
region.They generate a steady, eastward, equatorialcurrent both
east and west of the forcing region: the bounded YJ.
Eastern-boundary reflections
For low frequencies, the incoming Kelvin wave reflects as a packet
of Rossby waves (Moore, 1968).with the waves corresponding to
larger values propagating offshore more slowly. 1 3 5 7 Movie G1
The zonal current of the Kelvin wave divides at the Rossby-wave
front to flow along the edges of the wave packet. Adjustment to
steady state
d (1 month) In response to forcing by a patch of wind in the
interior ocean, KWs reflect from the eastern boundary as a packet
of RWs creating a characteristic wedge-shaped pattern.In addition,
wind-generated RWs reflect from the western boundary to return to
the interior ocean. Rossby wave Kelvin wave d (6 months) Coastal KW
Rossby wave Kelvin wave Equatorial jet d (1 year) Reflected Rossby
waves After multiple reflections, the solution eventually adjusts
to a state of Sverdrup balance. In response to an annual or
semiannual oscillation of the equatorial winds, these adjustments
continue to happen throughout the year in the Indian Ocean. Rossby
wave d (5 years) Reflected Rossby waves Near Sverdrup flow
Eastern-boundary reflections
Remarkably, the characteristic wedge shape and westward propagation
is visible in satellite data.The figure shows global maps of
filtered sea level from TOPEX/Poseidon on April 13 and July 31, It
shows a Rossby-wave packet generated by the reflection of an
equatorialKelvin wave forced by intraseasonal winds in the western
ocean. (After Chelton and Schlax, 1996.) Movies F In Movies F, for
zonal winds Kelvin and Rossby waves radiate from the wind patch
leaving behind a Sverdrup balanced flow + a bounded Yoshida Jet.
For meridional winds, Yanai and Rossby waves radiate from the wind
path to establish a Sverdrup circulation.There is no bounded
Yoshida Jet because meridional winds cannot generate one (wrong
symmetry). Fedorov and Brown, 2007 Multi-baroclinic mode
adjustment
How does the LCS model adjust when many baroclinic modes are
included? d (1 month) The plot shows the n = 1 responsewithout
damping. It also illustrates the n > 1 responses, except that
thecurrents are narrower in y (Rn < R1) and the wave speeds are
smaller (cn < c1). Rossby wave Kelvin wave d (6 months) Coastal
KW Rossby wave Kelvin wave With damping, the n > 1 responses are
increasingly damped since = A/cn2.In that case, waves that radiate
from the forcing region are weakened for larger n.For sufficiently
large n, then, the response is confined to the forcing region.
Equatorial jet d (1 year) Reflected Rossby waves See McCreary
(1981) for a detailed description of how the structure changes with
n. Rossby wave d (5 years) Reflected Rossby waves Near Sverdrup
flow Multi-baroclinic mode adjustment
When the LCS model includes damping (vertical mixing), a realistic
steady flow field is produced near the equator. EUC McCreary (1981)
Movies I1a, I1b & I1c Solutions for periodic winds Evanescent
waves /o k/o
As for the coastal model, there are two wavenumbers, k1,2,
associatedwith each value.The wavenumbers k1 (k2) describe waves
with eastward (westward) group velocity or decay. /o k/o 1 3 Also
as for the coastal model, the wavenumbers are real for small
(Rossby waves) and become complex as increases. Eventually, they
become real again for even larger (gravity waves). The region of
complex roots for = 1 waves is indicated by the shading.Such waves
exist only along boundaries, where they superpose to generate
-planecoastal Kelvin waves. Movies G2 and G3 Critical frequencies
/o k/o
Free Rossby waves exist at frequencies only below the shaded
region. Consider how the ocean responds to oscillatory winds at
different frequencies.At frequency 1, the wind can only excite
Kelvin and Yanai waves.At frequency 2, it can also excite = 1
RWs.At frequency 3,it can also excite = 1, 2, and 3 RWs. /o k/o 1 3
Movies G2 and G3 Vertical propagation Recall that the vertical
structure ofwaves in the LCS model satisfy Rather than to look for
solutions as expansions in vertical modes, n(z), another way of
studying solutions to the LCS model is to look for approximate
solutions of the form, under the restriction that the background
stratification, Nb(z) varies slowly with respect to the vertical
wavelength of the wave, m(z) (the WKB approximation). In that case,
Since cn > 0, the replacement in the last equation already
assumes that m > 0.if m < 0, then the replacement should be
cn = Nb/m.So, the logic of these slides is a bit wrong, as they
dont correctly state where the assumption that m > 0 is
made.[NOTE: To fix this problem, I replaced m with |m| in the last
equation on this page and the top one on the next page.] and cn can
be replaced by Vertical propagation (KW beams)
With this change, the dispersion relation for equatorial Kelvin
waves is Group theory states that a packet of Kelvin waves (that
is, a superposition of several waves associated with different k
and m values) propagates at the group velocity Thus, the energy of
the packet propagates to the east with the slope Along a northern
coast, = cnk, and the signs of all the propagations are reversed.
Similar results hold along north-south oriented boundaries, with
the complication that Kelvin waves associated with a particular
baroclinic mode dont exist equatorward of the critical latitude.
The movies show Kelvin beams at several different frequencies,
radiating downward along beam paths. So, if phase propagates
upwards (m > 0), energy propagates downwards, and vice versa.
Vertical propagation (YW beams)
The dispersion relation for Yanai waves becomes Group theory states
that a packet of Yanai waves (that is, a superposition of several
waves associated with different k and m values) propagates at the
group velocity Along a northern coast, = cnk, and the signs of all
the propagations are reversed. Similar results hold along
north-south oriented boundaries, with the complication that Kelvin
waves associated with a particular baroclinic mode dont exist
equatorward of the critical latitude. The movies show Kelvin beams
at several different frequencies, radiating downward along beam
paths. Thus, the energy of the packet propagates to the east with
the slope the same slope as for Kelvin waves! Vertical propagation
(long-wavelength RWs)
For the RW dispersion curves, as tends to zero so does k. So, in
the low-frequency limit the RW disp. curves are non-dispersive.
This limit is known as the long-wavelength approximation. /o k/o 1
3 In this limit, RWs propagate vertically with a slope with a
steeper slope, and in the opposite direction from, KW and YWs.
Single baroclinic mode response
d (1 month) In response to forcing by a patch of switched-on,
easterly winds, Kelvin and Rossby waves radiate from the forcing
region, reflect from basin boundaries, and eventually adjust the
system to a state of Sverdrup balance. Rossby wave Kelvin wave d (6
months) If the wind oscillates, say, at the annual, semiannual, or
intraseasonal periods, waves are continuously generated. Equatorial
KWs and RWs continuously radiate from the forcing region.Coastal
KWs radiate around the perimeter of the basin, and eastern-boundary
RWs radiate into the interior ocean. Equatorial jet d (1 year)
Reflected Rossby-wave packet In response to an annual or semiannual
oscillation of the equatorial winds, these adjustments continue to
happen throughout the year in the Indian Ocean. d (5 years)
Sverdrup flow Multi-baroclinic mode response
Recall that when the LCS model includes vertical mixing (damping),
a realistic steady flow field is produced near the equator.What
happens if the wind oscillates, say, at annual, semiannual, or
intraseasonal periods? EUC The equatorial Indian Ocean is dominated
by vertically propagating equatorially trapped waves.Indeed, there
are virtually no steady flows at all! McCreary (1981) Without
damping, waves radiate from the forcing region along beams that
extend into the deep ocean and exhibit upward phase
propagation.Yanai and Kelvin beams extend downward and eastward,
and Rossby waves extend downward and westward. With damping, the
beams weaken away from the forcing region. Tropical instability
waves
Legeckis (1977, Science) first reported the presence of TIWs in the
eastern, tropical Pacific.TIWs were soon shown to have a large
impact on the momentum and heat fluxes in the region.Philander
(1976, 1978, JGR) argued that TIWs were caused by barotropic
instability.Yu et al. (1992, Prog. Oceanogr.) later suggested that
an instability of the temperature front was involved.Luther and
Johnson (1990) suggested that there was more than one type of TIWs.
1) Note that clouds seem to be following the TIW front in the
northern hemisphere.Modeling work in the past decade has explored
the impact of TIWs on the atmosphere, using both AGCMs and CGCMs.
Similar TIWs were soon observed in the Atlantic Ocean.Their
dynamics are essentially the same as for the Pacific TIWs. Tropical
instability waves
Cox, M.D., 1980: Generation and propagation of 30-day waves in a
numerical model of the Pacific. J. Phys. Oceanogr., 10, 11681186.
The above solution assumes that Nb is constant.When Nb weakens with
depth, as it does in the real ocean, ray paths slope more steeply
with depth. Michael Cox (1980) reported a Yanai-wave beam forced by
surface TIWs in his OGCM solution.Ascani & coworkers (2009)
explored the idea that deep equatorial currents are caused by an
instability of the Yanai-wave beam generated by TIWs.To simulate
the effect of TIWs, they forced their OGCM by a wind stress with
the wavelength (~1000 km) and period (~30 days) of a typical TIW,
generating the Yanai-wave beam shown above. Tropical instability
waves
Harvey and Patzert (1976) likely detected the off-equatorial u
field of the TIW-driven Yanai beam on the ocean bottom east of the
Galapagos.(Sadly the mooring on the other side of the equator
failed.) Upward phase propagation in the EEIO
Masumoto et al. (2005) The u field (b & d) shows a strong
semiannual cycle. Above 200 m, the phase of upropagates upwards,
indicating that it is remotely forced (wave) signal! There is a
difference in the signals visible in u and v, with the latter
exhibiting strength in the 1020-day band. Note the upward phase
propagation of the semiannual signal, a property that the LCS model
can easily simulate. Movies J Bounded (3-d) Yoshida Jet
If the pressure-gradient term pnx is then included, the flow field
has both a realistic amplitude and structure. If only the damping
term (A/cn2)un is included in the zonal momentum equation, the flow
stops accelerating, but it is unrealistically fast (the unit is
km/s) and extends to the bottom. Zonal flows along the equator dont
continue to accelerate in reality or models.Why not?
Eastern-boundary reflections (Moores chain rule)
Suppose the ocean is forced by a patch of oscillating zonal wind
confined to the interior ocean.It generates an equatorial Kelvin
wave, that radiates to the eastern boundary of the basin. There can
be no zonal flow through the boundary.How does the system adjust to
prevent this flow? Dennis Moore showed that a packet of equatorial
waves with the zonal velocity field, (7) are generated at the
eastern boundary.In (7), the wavenumbers k1 correspond to waves
with westward group velocity or decay. Eastern-boundary reflections
(Moores chain rule)
For convenience, let the eastern boundary be located at x = 0.The
uK field there is and it must be cancelled by To eliminate uK, we
use the u1 wave and set B1 = Uo.With this choice the 0 terms are
cancelled, but a 2 term is created.We use the u3 wave to cancel
this term, and so on. In general, once B is known then the
recursion relation for Moores famous chain rule. Long-wavelength
approximation
Equations for the un, vn, and pn for a single baroclinic mode are
In the equatorial region, there are no simplifications that allow
for mathematically simple solutions (i.e., that allow y-derivatives
to be dropped. One useful simplification (analogous to the coastal
one) is to adopt the long-wavelength approximation, which restricts
the zonal flow to be in geostrophic balance.For convenience, also
drop horizontal mixing from the x-momentum equation (but that is
not necessary).