A Didactic Approach to Linear Waves in the Ocean

0.18 MB·English

by  F. J. Beron-Vera

A Didactic Approach to Linear Waves in the Ocean
F. J. Beron-Veraa)
RSMAS/AMP, University of Miami, Miami, FL 33149
(Dated: February 2, 2008)
The general equations of motion for ocean dynamics are presented and the waves supported
by the (inviscid, unforced) linearized system with respect to a state of rest are derived. The
linearizeddynamicssustainsonezerofrequencymode(calledbuoyancymode)inwhichsalinityand
temperaturerearrangeinsuchawaythatseawaterdensitydoesnotchange. Fivenonzerofrequency
4 modes (two acoustic modes, two inertia–gravity or Poincar´e modes, and one planetary or Rossby
0 mode)arealsosustainedbythelinearizeddynamics,whichsatisfyanasymptoticgeneraldispersion
0
relation. Themostusualapproximationsmadeinphysicaloceanography(namelyincompressibility,
2
Boussinesq, hydrostatic, and quasigeostrophic) are also consider, and their implications in the

n reduction of degrees of freedom (number of independent dynamical fields or prognostic equations)
a of, and compatible waves with, the linearized governing equations are particularly discussed and
J emphasized.

9
PACSnumbers: 43.30.Bp,43.30.Cq,43.30.Ft

]
h
p I. INTRODUCTION II. GENERAL EQUATIONS OF MOTION
-
d
e The goal of this educational work is to show how the Let x := (x,y) be the horizontal position, i.e.
.
s various types of linear waves in the ocean (acoustic, tangential to the Earth’s surface, with x and y its
c inertia–gravity or Poincar´e, and planetary or Rossby eastward and northward components, respectively; let z
i
s waves)canbeobtainedfromageneraldispersionrelation be the upward coordinate; and let t be time. Unless
y inanapproximate(asymptotic)sense. Knowledgeofthe otherwise stated all variables are functions of (x,z,t) in
h
theoryofpartialdifferentialequations,andbasicclassical this paper.
p
[ and fluid mechanics are only needed for the reader to Thethermodynamicstateoftheoceanisdeterminedby
understand the material presented here, which could be three variables, typically S (salinity), T (temperature),
1 taught as a special topic in a course of fluid mechanics and p (pressure, referred to one atmosphere). Seawater
v
for physicists. density, ρ, is a function of these three variables, i.e. ρ=
5
ρ(S,T,p), known as the state equation of seawater. In
3 The exposition starts by presenting the general
particular,
0 equations of motion for ocean dynamics in Sec. II. This
1 presentation is not intended to be rigorous, but rather ρ−1Dρ=α DS α DT +α Dp. (1)
0 S − T p
conceptual. Accordingly, the equations of motion are
4 Here, D := ∂ + u ∇ + w∂ is the substantial or
0 simplifiedasmuchaspossiblefordidacticpurposes. The materialderivattive,wh·ereuandwz arethehorizontaland
/ general dispersion relation for the waves supported by
s verticalcomponentsofthevelocityfield,respectively,and
the (inviscid, unforced)linearizeddynamics with respect
c ∇ denotes the horizontal gradient; α := ρ−1(∂ ρ)
i to a quiescent state is then derived in Sec. III. This is S S T,p
s and α := ρ−1(∂ ρ) are the haline contraction and
y donebyperformingaseparationofvariablesbetweenthe T T S,p
thermal expansion coefficients, respectively; and α :=
h verticaldirection,ononeside,andthehorizontalposition ρ−1(∂ ρ) = α Γ+ρ−1c−2, where Γ is the adiabpatic
p and time, on the other side. Sec. IV discusses the p T,S T s
: implications of the most common approximations made gradientandcs is the speed ofsound,whichcharacterize
v
in oceanography (namely incompressibility, Boussinesq, the compressibility of seawater.
i
X hydrostatic, and quasigeostrophic) in the reduction of The physical state of the ocean is determined at every
r degrees of freedom (number of independent dynamical instant by the above three variables (S,T,p) and the
a fields or prognostic equations) of, and compatible waves three components of the velocity field (u,w), i.e. six
with, the linearized governing equations. Particular independent scalar variables. The evolution of these
emphasis is made on this important issue, which is variables are controlled by
vaguelycoveredinstandardtextbooks (e.g. Refs. 2,3,5).
DS =F , (2a)
S
Someproblemshavebeeninterspersedwithinthetextto
DT =ΓDp+F , (2b)
helpthereadertoassimilatethematerialpresented. The T
solutions to some of these problems are outlined in App. Dp=−α−p1(∇·u+∂zw+αTFT −αSFS), (2c)
A. Du= ρ−1∇p+Fu, (2d)
−
Dw= ρ−1∂ p g+F . (2e)
z w
− −
In Newton’s horizontal equation (2d), the term Fu
a)Electronicmail: [email protected] represents the acceleration due to the horizontal
Typeset by REVTEX
2
components of the Coriolis and frictional forces. In Here, ˆz is the upward unit vector and f := 2Ωsinϑ,
Newton’s vertical equation (2d), the term F represents where Ω is the (assumed constant) spinning rate of the
w
the acceleration due to the vertical component of the Eartharound its axis and ϑ is the geographicallatitude,
Coriolis and frictional forces, and g is the (constant) is the Coriolis parameter. For simplicity, we will avoid
acceleration due to gravity. The term F in the salinity working in full spherical geometry. Instead, we will
S
equation (2a) represents diffusive processes. The term consider f = f +βy, where f := 2Ωsinϑ and β :=
0 0 0
F in the thermal energy equation (2b), which follows 2ΩR−1cosϑ with ϑ a fixed latitude and R the mean
T 0 0
from the first principle of thermodynamics, represents radius of the planet, and ∇ = (∂ ,∂ ), which is known
x y
the exchange of heat by conduction and radiation, as as the β-plane approximation. It should remain clear,
wellasheating bychangeofphase,chemicalreactionsor however, that a consistent β-plane approximation must
viscous dissipation. The pressure or continuity equation include some geometric (non-Cartesian) terms.7 Neither
(2c) follows from (1). these terms nor those of the Coriolis force due to the
horizontal component of the Earth’s rotation contribute
Problem 1 Investigate why (2d,e) do not include the toaddwavestothelinearizedequationsofmotion. Their
centrifugal force which would also be needed to describe neglection is thus well justified for the purposes of this
thedynamicsinanoninertialreferenceframesuchasone paper.
attached to the Earth.
Since adiabatic compression does not have important
III. WAVES OF THE LINEARIZED DYNAMICS
dynamical effects, in physical oceanography it is
commonly neglected. This is accomplished upon
Consider a state of rest (u = 0, w = 0) characterized
introducing the potential temperature θ, which satisfies
by dp /dz = ρ g, where p (z) and ρ (z) are reference
αθDθ =αT(DT ΓDp), so that ρ=ρ(S,θ,p) and (1) is r − r r r
− profiles of pressure and density, respectively. Equations
consistently replaced by
(4) and (6), linearized with respect to that state, can be
ρ−1Dρ=α DS α Dθ+ρ−1c−2Dp, (3) written as
S − θ s
′ ′
where here it must be understood that α = (∂ ρ) ∂ ζ =w , (7a)
S S θ,p t
and αT = (∂θρ)S,p. Equations (2b,c) then are replaced, ∂ p′= ρ c2 ∇ u′+∂−w′ , (7b)
respectively, by t − r s · z
∂tu′=−ρ−r 1∇(cid:0)p′−fˆz×u′, (cid:1) (7c)
Dθ=Fθ, (4a) ∂tw′=−ρ−r 1∂z+p′−N2ζ′. (7d)
Dp= ρc2(∇ u+∂ w+α F α F ). (4b)
− s · z θ θ − S S Here, primed quantities denote perturbations with
As our interest is in the waves sustained by the respect to the state of rest; ∂± := ∂ gc−2; N2(z) :=
linearized dynamics, we do not need to consider either g(ρ−1dρ /dz + gc−2) is thez squarze±of tshe reference
− r r s
diffusive processes or allow the motion to depart from Brunt-Va¨is¨al¨afrequency; and c is assumed constant. In
s
isentropic. Hence, we will set FS 0 Fθ so that addition to the above equations, it is clear that
≡ ≡
equations (2a) and (4) can be substituted, respectively,
′
by ∂ S = wdS /dz, (8a)
t r
−
′
∂ θ = wdθ /dz, (8b)
Dζ =w, (5a) t − r
Dp=−ρc2s(∇·u+∂zw), (5b) whereSr(z)andθr(z)arereferencesalinityandpotential
density profiles, respectively.
whereζ istheverticaldisplacementofanisopycnalwhich
is defined such that ρ=ρ (z ζ).
r − Problem 3 Work out the linearization of the equations
Problem 2 Showthatequations(2a)and(4a)certainly of motion.
lead to (5a) when F 0 F .
S θ
≡ ≡
Wewillalsoneglectfrictionaleffects,sothatequations A. Zero Frequency Mode
(2d,e) are seen to be nothing but Euler equations of
(ideal) fluid mechanics with the addition of the Coriolis The linearized dynamics supports a solution with
force. The latter will be further considered as due solely vanishing frequency (∂ 0) such that
t
to the vertical component of the Earth rotation. Thus, ≡
the following simplified form of equations (2d,e) will be ζ′ 0, p′ 0, u′ 0, w′ 0, (9)
considered: ≡ ≡ ≡ ≡
as it follows from (7), but with
Du= ρ−1∇p fˆz u, (6a)
− − ×
Dw= ρ−1∂ p g. (6b) S′ =0, θ′ =0, (10)
z
− − 6 6
3
as can be inferred from (8). Namely for this solution Equation (16) can be presented in different forms
thesalinityandtemperaturefieldsvarywithoutchanging according to the approximation performed. Under
the density of the fluid. More precisely, one has, on one the incompressibility approximation, which consists of
hand, ρ′ = ρ (g−1N2+gc−2)ζ′, and, on the other, ρ′ = makingthe replacement∂ p′+gw′ 0inthe continuity
ρ (α S′ αrθ′ +α p′), wshere the α’s are evaluated at equation (7b), equation (1t6) takes t7→he form
r S − θ p
the reference state. By virtue of (9) then it follows that
∂+(ρ ∂ F)+ρ c−2 N2 ω2 F =0. (18)
′ ′ z r z r −
α S α θ =0. (11)
S θ
− This approximation correspo(cid:0)nds forma(cid:1)lly to taking the
This so-called buoyancy mode describes small scale limit c−2 0. The hydrostatic approximation, in turn,
processes in the ocean such as double diffusion. consistssof→makingthereplacement∂ w′ 0inNewton’s
t
7→
vertical equation (7d). This way, without the need
of assuming any particular temporal dependence, it
B. Nonzero Frequency Modes followsthatζ′ = pc∂+(ρ ∂−F)/(ρ N2). Consequently,
− z r z r
equation (16) reduces to
Upon eliminating w between (7a) and (7d), and
∂+ ρ ∂−F +ρ c−2 c−2 N2F =0. (19)
proposing a separation of variables between z, on one z r z r − s
side, and (x,t), on the other side, for the horizontal
This approx(cid:0)imation(cid:1)is val(cid:0)id for ω2 (cid:1) N2, i.e. periods
velocity and pressure fields in the form ≪
exceedingthe localbuoyancyperiodwhichtypicallyisof
u′ =uc(x,t)∂−F(z), p′ =ρ (z)pc(x,t)∂−F(z), (12) about 1 h. (Of course, this approximation implies that
z r z
of incompressibility as it filters out the acoustic modes
it follows that whose frequencies are much higher than the Brunt-
Va¨is¨al¨a frequency.) Another common approximation is
c−s 2∂∂z−ζ′F+∂tρp−c1+∂∂+t(cid:0)u∂cz−ρ+F∂∇f−ˆzF·×upcuc+c++∂Nz−∇tζ2pζ′(cid:1)c′===000.,, (((111333bac))) ttUhhneedBreeropultashsciiesnmeasepqnptarsopρxprirmo7→xaitmiρ¯oan=ti,ocneoq,nuwsatht.iiocahnndctoa∂nkzs±eiss7→ttsho∂efzmsiinmakp(i7lne)gr.
tt r z r z form
Nfoormw,ea−sisωutm, firnogma(1c3ocm)mon(cid:0)oenotbetmapi(cid:1)nosral dependence of the d2F/dz2+ c−2−c−s 2 N2−ω2 F =0. (20)
ζ′ = pc∂z+(ρr∂z−F). (14) Pverroybgloeomd5forShthowe(cid:0)otcheaatnthbeut(cid:1)B(cid:0)nouotsssinoefsoqr(cid:1)atphperoaxtmimoastpihoenreis.
− ρ (N2 ω2)
r − Hint: This approximationrequires c2 gH, where H is
s ≫
Then, upon substituting (14) in (13a) it follows that a typical vertical length scale.
1 ∂+(ρ ∂−F) ∇ uc
c−2 ∂− z r z = · =c−2, (15)
s − ∂z−F z (cid:20)ρr(N2−ω2)(cid:21) − ∂tpc 1. Horizontal Structure
where c is a constant known as the separation constant. To describe these waves is convenient to introduce a
Clearly, cs must be chosen as a constant in order for potential ϕ(x,t) such that6,9
the separation of variables to be possible. From (15) it
follows, on one hand, that pc= c2(∂ +f∂ )ϕ, (21a)
ty x
−
∂+ ρ ∂−F +ρ N2 ω2 c−2 c−2 F =0, (16) uc= c2∂xy+f∂t ϕ, (21b)
z r z r − − s vc=(cid:0)∂tt c2∂xx (cid:1)ϕ, (21c)
and ta(cid:0)king in(cid:1)to acc(cid:0)ount (13b(cid:1))(cid:0), it follows(cid:1), on the other −
hand, that which allows one to(cid:0)reduce sys(cid:1)tem (17) to a single
equation in one variable:
∂ pc+c2∇ uc=0, (17a)
t
∂tuc+fˆz uc+∇· pc=0. (17b) Lϕ:= ∂t ∂tt+f2(y)−c2∇2 −βc2∂x ϕ=0. (22)
×
The linea(cid:8)r d(cid:2)ifferential operator(cid:3) conta(cid:9)ins a variable
Equation (16) governs the vertical structure of the L
coefficient and, hence, a solution to (22) must be of the
perturbations,whereassystem(17)controlstheevolution
form
of their horizontal structure.
ϕ=Φ(y)ei(kx−ωt) (23)
Problem 4 Show that the alternate separation of
variables which uses F(z) instead of ∂z−F(z) leads to with Φ(y) satisfying
a vertical structure equation with a singularity where
ω2 N2. d2Φ/dy2+l2(y)Φ=0 (24)
≡
4
where Equation(30) can be understood in two different senses.
Within the realm of the WKB approximation, (31)
k ω2 f2(y)
l2(y):= k2 β + − . (25) defines a local vertical wavenumber, and a solution to
− − ω c2 (30) oscillates like
Now, if l2(y) is positive and sufficiently large, then Φ(y) F(z) e±i zdzm(z). (32)
oscillates like ∼
TheothersenseisthatofvertRicalnormalmodes,inwhich
Φ(y) e±i ydyl(y). (26) (30) is solved in the whole water column with boundary
∼ conditions
R
This is known as the WKB approximation (cf. e.g. Ref.
F( H)=0, (33a)
4), where l(y) defines a local meridional wavenumber in −
gF(0)=c2dF(0)/dz. (33b)
the approximate (asymptotic) dispersion relation
Condition(33a)comes fromimposing w′ =0 atz = H
k −
ω2 (f2+c2k2) β =0, (27) where H, which must be a constant, is the depth of the
− − ω fluid in the reference state. Condition (33b) comes from
the fact that p′ = gζ′ at z = 0, which means that the
where k:=(k,l) is the horizontal wavenumber.
surface is isopycnic (i.e. the density does not change
System (17) also supports a type of nondispersive
on the surface). This way one is left with a classic
waves called Kelvin waves. These waves have vc 0
≡ Sturm–Liouville problem. Making the incompressibility
and thus are seen to satisfy
approximation and assuming a uniform stratification in
the reference state, namely N = N¯ = const., it follows
∂ pc+c2∂ uc=0, (28a)
t x that
∂ uc+∂ pc=0, (28b)
t x
ω2 =N¯2 mgtanmH. (34)
fuc+∂ypc=0. (28c) −
(Notice that to obtain m is necessary to fix a value of
Clearly,these wavespropagateasnondispersivewavesin ω.)Inthehydrostaticlimit, ω2 N¯2,itfollowsthatthe
the zonal (east–west) direction—as if it were f 0— vertical normal modes result fro≪m
≡
and are in geostrophic balance between the Coriolis and
tanmH =s/(mH), (35)
pressure gradient forces in the meridional (south–north)
direction. From (28a, b) it follows that where s := N¯2H/g, which is a measure of the
stratification, is such that 0<s < by static stability.
pc =A(y)K(x ct) cuc, (29) In the ocean s is typically very sma∞ll, so for s 1 from
− ≡ ≪
(35) it follows that
where K() is an arbitrary function. By virtue of (28c)
then it fol·lows dA/dy+fA/c=0, whose solution is N¯/√gH if i=0,
m = (36)
i iπ/H if i=1,2, .
A(y) e− ydyf(y)/c =e−(f0y+12βy2)/c, The firstmode is(cid:26)calledthe external or b·a·r·otropic mode;
∝
R therestofthemodesaretermedtheinternal orbaroclinic
which requires, except there where f 0 (i.e. the
0 modes, which are well separated from the latter in what
≡
equator), the presence of a zonal coast to be physically length scale respects. More precisely, the Rossby radii
meaningful. of deformation are defined by R := N¯/(m f ); for
i j 0
| |
the barotropic mode R = √gH/ f whereas for the
0 0
Problem 6 Consider the Kelvin waves in the so-called baroclinic modes R = N¯H/(iπ f|) | √sR /(iπ)
i 0 0
f plane, i.e. with β 0. R . Finally, the rigid lid appr|oxi|ma≡tion consists ≪of
≡ 0
making w′ = 0 at z = 0, which formally corresponds to
takethelimitg in(33b). Thisapproximationfilters
→∞
2. Vertical Structure out the barotropic mode since it leads to tanmH =0.
Under the Boussinesq approximation the five fields Problem 7 Demonstrate that p′ =gζ′ at z =0.
of system (7) remain independent, thereby removing no
wave solutions. We can thus safely consider the vertical
3. General Dispersion Relation
structure equation (20), which we rewrite in the form
d2F/dF2+m2(z)F =0 (30) Upon eliminating c between (25) and (31) it follows
that
where
k2+βk/ω m2
= +c−2, (37)
m2(z):= N2(z) ω2 c−2 c−2 . (31) ω2 f2 N2 ω2 s
− − s − −
(cid:2) (cid:3)(cid:0) (cid:1)
5
which is a fifth-order polynomial in ω that constitutes
the general dispersion relation for linear ocean waves in ∂tζ′ =···
athnisapsyampeprt.oAticppWroKxiBmasteensreo.otTshofis(3is7)thaerem: ain result of ∂∂ttup′′==······ :5 independent fields ↔5 waves: 221 APRWWW
acoustic:ω2 =(k2+m2)c2, (38) ∂tw′ =···
s
which holds for ω2 ≫N2 (i.e. very high frequencies); incompressibility ↓ ∂zw′ =−∇·u′ 
∆p′=−∇·fˆz×u′−∂z(N2ζ′)
k2N2+m2f2
Poincar´e:ω2 = , (39)
k2+m2 ∂tζ′ =··· 2 PW
:3 independent fields ↔3 waves:
which follows upon taking the limit c−2 0 and is valid ∂tu′ =··· ) ( 1 RW
s →
for frequencies in the range f2 <ω2 <N2; and
quasigeostrophy ↓ ∂zu′ = N2ˆz×∇ζ′
βk f
Rossby:ω = , (40) 0
−k2+(m2/N2)f2
∂tq′ =···:1 independentfield ↔1 wave : 1 RW
which also follows in the limit c−2 0 but is valid for
ω2 f2 (i.e. very low frequenciess).→
≪
TABLE I: Reduction of independent fields (and, hence,
Problem 8 Demonstrate that the classical dispersion prognostic equations) by the incompressibility and quasi-
relationsforPoincar´ewaves,ω2 =f2+c2k2,andsurface geostrophic approximations. Here, AW, PW, and RW
gravity waves, ω2 =g k tanh k H, are limiting cases of stand for acosutic waves, Poincar´e waves, and Rossby wave,
(39). | | | | respectively; ∆ := ∇2 + ∂zz is the three-dimensional
Laplacian; and q′ := f + ∇2p′/f0 + ∂z(f0N−2∂zp′) is the
so-called quasigeostrophic potential vorticity.
IV. DISCUSSION
The inviscid, unforced linearized equations of motion As a consequence, the two AW are filtered out and
(7) have five prognostic equations for five independent one is left with the two PW and the RW. With
dynamical fields. As a consequence, five is the these two approximations the Euler equations (7)
number of waves sustained by (7) which satisfy the reduces to the so called Euler-Boussinesq equations.
general dispersion relation (37) in an asymptotic WKB Performing in addition the hydrostatic approximation,
sense. In proper limits, two acoustic waves (AW), whichcorrespondstoneglecting∂ w′inNewton’svertical
t
two Poincar´e waves (PW), and one Rossby wave (RW) equation, does not amount to a reduction of degrees of
can be identified. The fact that the number of waves freedombecausetheverticalvelocityisalreadydiagnosed
supportedby the linearizeddynamics equalsthe number by the horizontal velocity. In this case, the density field
of independent dynamical fields or prognostic equations, diagnoses the pressure field through
i.e. the degrees of freedom of (7), means that the waves
constitute a complete set of solutions of the linearized ∂ p′ = N2ζ′. (43)
z
dynamics (cf. Table I). −
The number of possible eigensolutions can be reduced
With this approximation (which implies that of incom-
is approximations that eliminate some of the prognostic
pressibility) and the Boussinesq approximation, system
equations,orindependentdynamicalfields,ofthesystem
(7) reduces to what is known in geophysical fluid
are performed. The Boussinesq approximation, which
dynamics as the primitive equations. Finally, one
is very appropriate for the ocean, does not eliminate
approximation that eliminates independent fields is the
prognostic equations and has the virtue of reducing
quasigeostrophic approximation, which is often used to
the mathematical complexity of the governing equations
studylowfrequencymotionsintheocean,andtheEarth
considerably. The incompressibility approximation, in
and planetary atmospheres. In this approximation the
turn, removes two degrees of freedom: the vertical
density diagnoses the horizontal velocity through the
velocity is diagnosed by the horizontal velocity,
“thermal wind balance,”
∂ w′ = ∇ u′, (41)
z
− ·
and the latter along with the density diagnose the ∂ u′ = N2ˆz ∇ζ′, (44)
z
f ×
pressure field through the three-dimensional Poisson 0
equation
thereby removing two degrees of freedom and leaving
( 2+∂ )p′ = ∇ fˆz u′ ∂ (N2ζ′). (42) only one RW.
zz z
∇ − · × −
6
Acknowledgments continuity equation (7b) readily follows upon noticing
that,uptoO(a),c−2Dp=c−2(∂ p′ ρ gw′).UptoO(a),
mocaeTtaehnreoiagalruattophhosyrtuhadatesnCitmsICpoafErtStheEedd(leEoccnttusoerrneasaldbpaar,soegBdraaojmnatiChneaplpihfroyerssniecinaatl, a∂Dntρζd′−−Dcζ−sw2−′D.wpTh==en∂0tfρirt′os−mfogllt−oh1weρsrrNteshla2awttio′ρt−n′s=ch−−sipc2s−s∂rt2Dpp′ρ′a+−ndgc−−sD12ζρDr−pNw=2ζ=′0.
Mexico). Part of this material is inspired on a seminal Bearing in mind the latter relation and the fact that
homeworkassignedbythelateProfessorPedroRipa. To dpr/dz =−gρr,theO(a)verticalNewton’sequation(7d)
his memory this article is dedicated. then follows.
Problem 4 For the ocean c 1500 m s−1 √gH
APPENDIX A: SOLUTIONS TO SOME OF THE 200 m s−1; by contrast, for tsh∼e atmosphere c≫ 350 m∼
PROBLEMS s
s−1 √gH with H 12 km, which is the typic∼alheight
∼ ∼
of the troposphere.
Problem 1 To describe the dynamics in a noninertial
reference frame such as one tied to the rotating Earth,
twoforcesmustbeincluded: the Coriolisandcentrifugal Problem 7 At the surface z =η it is w =∂ η+u ∇η
t
forces. However, Laplace1 showed that if the upward andp=0(here,pisakinematicpressure,i.e. divide·dby
coordinate z is chosen not to lie in the direction of the a constant reference density ρ¯). Writing η = η′+O(a2)
gravitational attraction, but rather to be slightly tilted and Taylor expanding about z = 0 it follows, on one
towardthenearestpole,thecentrifugalandgravitational hand,
forcescanbemadetobalanceoneanotherinahorizontal
plane(cf. alsoRef. 8). WiththischoicetheCoriolisforce w′+η′∂ w′+O(a3)=∂ η′+u′ ∇η′+O(a3) (A2)
z t
is the only one needed to describe the dynamics. Notice ·
thattheabsenceofthecentrifugalforceinasystemfixed
at z =0, and, on the other hand,
to the Earthis whatactuallymakesrotationeffects real:
they cannot be removed by a change of coordinates. p +(p +dp /dz)η′+η′∂ p′+O(a3)=0 (A3)
r r r z
Problem 2 In the absence of diffusive processes, the at z =0. From (A2) it follows, to the lowest order, w′ =
(isuoζpy+cnˆzawlζz)]=·[∇ζ ζis−aˆzm(1at−er∂iazlζ)s]ur=fac0e.,Hi.eer.e,[uuζ++ˆzˆzww−ζ t∂htηen′ aittfzol=low0s. tShiantceη′w=′ =ζ′ a∂ttζz′,=fo0r.aTawkaivneg(iin.eto. a∂ctc6=oun0)t
denotes the velocity of some point on the surface [the
the latter and choosing ρ¯ = ρ (0), from (A3) it follows,
velocity of a surface is not defined and it only makes to the lowest order, p′ =gη′ rgζ′ at z =0 since p = 0
sense to speak of the velocity in a given direction, e.g. ≡ r
and dp /dz = gρ /ρ¯ g at z =0.
thenormaldirection,inwhosecaseitisˆz(1 ∂zζ) ζ]. r − r ≡−
− −∇
From the trivial relation z ζ = 0 then it follows that
(uζ +ˆzwζ) [∇ζ ˆz(1 ∂zζ)−]= ∂tζand,hence,Dζ =w Problem 8 The classical dispersion relation for
· − − −
at z =ζ. Poincar´e waves corresponds to the hydrostatic limit,
which requires m2 k2 (i.e. that the vertical length
Problem 3 To perform the linearization of the equa- scales be shorter t≫han the horizontal length scales).
tions of motion, we write Under this conditions, ω2 =f2+k2N2/m2 =f2+c2k2.
To obtain the dispersion relation for surface gravity
(u,w,ζ) = (u′,w′,ζ′) + , wavesoneneedstotakeintoaccountboundaryconditions
···
(p,ρ) = (ρr,pr) + (ρ′,p′) + ··· , (A1) (m323)=: makk2i,nagnNd,2o≡n0ot≡hefr,2tihtaftolωlo2w=s, onmognteanhmanHd,.tThhaet
O : 1 a a2 dispers−ion relation ω2 = g k tanh −k H then readily
| | | |
follows.
where a is an infinitesimal amplitude. The O(a)
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