Table Of ContentIntroduction 1
1 Equations and principles
of atmospheric large-scale dynamics
Introduction
The atmosphere contains a wide spectrum of motions ranging from microturbulence in the order of
centimeters to circulation patterns of global size (see Fig. 1). Speaking of large-scale motions means that
phenomena are considered that range horizontally from several hundred kilometers up to the circumference of
the earth (shaded area in Fig. 1). These large scales, also called “synoptic scales” are resolvable with the
meteorological routine observing network, they possessb y far the largest part of the whole energy and are most
relevant for weather and earth climate; these scales are therefore treated explicitly in numerical models of
weather prediction and of global climate. In this treatment, the physical effects of the smaller-scale phenomena
are included by functional relationships with the large-scale variables as parameters (the problem of
parameterization). The processest o be parameterized are in essencet he “turbulent” fluxes of momentum and
energy. Thus friction, for example, is in this context the convergence of the turbulent fluxes of momentum,
against which the molecular friction is negligibly small.
There exists a great number of textbooks on large-scale dynamics and thermodynamics of the atmosphere
(sees ection 1.6),w here one can obtain detailed information. It is not the purpose to write yet another book on
dynamic meteorology but rather to list up those equations and principles that are generally employed in large-
scale dynamics of the atmosphere.
Eddy
kinetic energy
Time scale -
Prediction periods
Short Medium tong Seasonal Climatic
range range range change change
1Omin lh 6h 2d 20d la 10a
I I .jlx^ ~~I I . I. .,.. I
/ / /
/ woves 1 / / /
Thunder- /
/ / /
’ 1
. ..-I
’ IU
Fig. 1. Time and spaces caleso f various atmosphericp henomena.T he shadeda rea on the upper right representst he
large scales,w hich are resolvableb y the observing network and which can explicitely be treated by global general
circulation modelsa nd weatherp rediction models.T he curve on the left representsth e spectrumo fatmospheric kinetic
energya s a function ofhorizontal scale;w ithin the hatchedp ortion the physicalp rocessecsa n generallyn ot be resolved,
one must seekt o expresst heir statistical effecti n terms of the large scalep arameters( the parameterizationp roblem)
(Basedu pon GARP PUBLICATION SERIES No 11 WMO Geneva 1973,p . 9).
Land&-Biimstein Fischer
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2 1.1 The fundamental laws and their mathematical representation
1.1 The fundamentall aws and their mathematical representation
1.1.1 Newton’s 2nd law and the equation of motion
Newton’s2 nd law statest hat in an inertial system,w hich is fixed relative to the stars, the masso f a particle
times the acceleration of the particle d,u,/dt equals the sum of the forces EGi acting on the particle. With
application to the earth’sa tmospherei t is commonly supposedt hat the particle consistso f a small volume of air
having unit mass.t hen
d u.
““=zGi [m.se2]=[N.kgw1],
dt
where F, denotes the “absolute velocity”. The forces per mass of meteorological interest are:
the pressureg radient force: - iVJp,
the gravitation force: ga,
the friction force: F,,
where p and Q are pressurea nd density of the air, respectively.
For conveniencet he atmospheric state is generally referredt o a coordinate systema ttached to the earth and
rotating at a constant angular velocity R; if u3d enotest he three-dimensionalv elocity field relative to the earth,
then one finds from u,=u3 +Q x r:
0, du,
-dr+2f2xu,+Rx(lRxr). (1.1)
dt
The first, seconda nd third term on the r.h.s.o f(l.1) are the relative acceleration,t he Coriolis accelerationa nd
the centripetal acceleration,r espectively.I t is customary to denote the latter two accelerationsw ith the inverse
sign as apparent forces, i.e. Coriolis force and centrifugal force.
Newton’s2 nd law appearst hen in a modified form called the equation ofmotion or the momentum equation:
du,
x=-e& I-2n xu,+g+F, [N.kg-‘1. (1.2)
The combined effect of the gravitation force and the centrifugal force is called gravity (per mass)g , i.e.
g=ga-Rx(f2xr).
Remark: The subscript 3 denotes“ three-dimensional” and has been used to distinguish the operators and
vectors from their two-dimensional horizontal form which has no corresponding subscript.
1.1.2 Mass conservation and continuity equation
The principle of the conservation of mass is expressedi n the continuity equation:
!!% +v,.u,=() [s-l]. (1.3a)
e dt
With the inclusion of water vapor as an active variable another continuity equation of the form
1
---+vyu,=g[s -l]
dew
(1.3b)
ew dt
is required wherep ,,.i s the density of water vapor and SWr epresentst he sourcesa nd sinks of water vapor (masso f
water vapor per volume and time).
Fischer
Land&BBmsrein
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1.1 The fundamental laws and their mathematical representation 3
1.1.3 The gas law and the equation of state
Though the atmosphere is a mixture of gases,i t obeys to a satisfactory degree the equation of state for a
perfect gas
p/@=RT [J.kg-‘1, (1.4)
where the gas constant R is a weighted averageo f the gas constants of the individual gaseso f the atmosphere. For
the composition of dry air which is constant below 100 km height one finds R = 287J kg- ’ K-r.
1.1.4 First law of thermodynamics
The equation of thermal energy states that the heat H added per unit mass and per unit time to a particle
will increase its internal energy (c,dT/dt) and partly will do work against the surroundings
Thus:
c$+p;; =H [W.kg-‘1. (1.5)
With the aid of (1.4) the first law of thermodynamics can also be written as
c drL!E=H CW.k-‘I,
p dt e dt
where use has been made from cp- c, = R.
For dry air, the specific heat at constant pressure is cp= 1004J kg-‘K-l, that of a constant volume is
c,=717Jkg-lK-‘.
Eq. (1.6) is the preferred form in meteorology. Defining the potential temperature as
@=T PO R’cp
(Poisson’se quation) [K] , (1.7)
0 P
where p0 is 1000h Pa, the first law of thermodynamics can also be expressedb y
T d@
c --=H [W.kg-‘1
‘0 dt
The potential temperature is a conservative quantity for an adiabatic process (H = 0).
1.1.5 The rate of change of kinetic energy
By scalar multiplication of (1.2) with ug one obtains the kinetic energy equation:
dK3 1
--~u3.v3p+v3.g-D3 CW.kg-'I,
dt
where K, =i(u,. n3)i s the kinetic energy per unit mass.T he r.h.s. contains the work per unit mass and unit time
done by the pressure gradient, the gravity and the friction. The latter is defined as (rate of) dissipation
-D,=v,.F,<O.
1.1.6 The rate of change of vorticity
Upon applying the curl-operator v3 x to (1.2) the three-dimensional vorticity equation is obtained:
$curlv,f(V3.u3)curln.-(curlu,.V,)u,=-V,: xV,p+curlF, cs-21 (1.10)
Fischer
Landolt-Biimstein
New Series V/h
4 1.1 The fundamental laws and their mathematical representation
with
curl u, = curl UJ+ 2f2
as the absolute vorticity.
In dynamic meteorology, the vertical component of curlu, is widely useda nd generally defined as vorticity.
From (1.4).( 1.8) and (1.10)t he conservation of potential vorticity for frictionless (F=O) and adiabatic (H=O)
conditions can be derived (Ertel’st heorem):
$(iV,O.curlv,) =O. (1.11)
1.1.6.1 The condition of barotropy
Atmospheresw hose state obeys the condition
V3i xV,p=O or V,TxV,p=O
are denoted as barotropic atmospheres.T hey have the property that the surfaceso f equal density, temperature,
potential temperature are parallel to each other, i.e. the density is a function of pressurea lone. If the above
conditions are fulfilled for all times the state is called autobarotropic. The autobarotropic atmosphere with
p=const. plays an essential role in explaining the pure hydrodynamical behaviour of the real baroclinic
atmosphere( for further information see 1.4.2.6a nd 1.4.5).
1.1.7 Vector identities
For writing the foregoing equations in other forms, the following vector identities can be applied referring to
arbitrary three-dimensionalv ectors A and B and a scalar b:
V,.(bB)=bV,.B+B.V,b
V,x(bB)=bV,xB-BxV,b
V,.(AxB)=B(V,xA)-A.(V,xB)
V,x(A x B)=A(V,.B)-B(V,.A)-(A.V,)B+(B.V,)A
V,(A~B)=(B~V,)A+(A~V,)B+Bx(V,xA)+Ax(V,xB)
(B.V,)B=+VJB.B)-Bx(V,xB)
(V,.A)B=(A.V,)B+BV,.A
V, xV,b=O
V,.V,x B=O
V,xV,xB=V,(V,.B)-V;B
d,B-dB
+RxB
dr -dr
dB aB
- - +(ux.V,)B
Xi- at
db ah
+ u3. V,b gradient-form expansion
dr=at
dB apB
ed, = at +(V,.eu,)B
db aeb
e~=~+V3.eu3b flux-form expansion
V, x B = curl B (three-dimensionalv orticity)
V, B- div B (divergence)
V,b = grad b (gradient)
V, .V3b-V$b (Laplace Operator).
Land&-Bbmctein
New Series v ‘43
1.2 Spherical coordinates 5
db ab
IS called the total time derivative, i.e. the temporal change of b following the motion; - is the local time
x at
derivative, i.e. the temporal change at a fixed point in space. - u3. V3b is defined as advection in meteorol-
ogy, V, ’ p,b is the divergenceo f the flux of b.
1.2 Spherical coordinates
The wind vector (like any other vector) may be written as
u,=iu+ju+kw,
where i, j, k are unit vectors pointing to the east, north, and upwards, respectively; u, v, w are the velocity
components.
d’ is denoted as zonal,
u=rcoscp3i
dv
u=r - is denoted as meridional, and
dt
dr
w= - as vertical velocity
dt
with 1 as longitude, rp as latitude and r as distance from the earth’sc entre.
The unit vectors are dependent on the coordinates I, cpa nd r in the following way:
a a a
ani = jsincp-kcoscp, -i =-i=O
acp ar
(2.1)
ik=isintp, &k= j; ;k=O.
1.2.1 The equations in spherical coordinates and the order of magnitude
of individual terms
For large-scalem otions (also called synoptic scalem otions) one presumesf rom observationsi n midlatitudes
the following characteristic orders of magnitude:
horizontal wind (u, u): 10m s- ’
vertical wind (w): lo-‘rns-’
horizontal length scale: lo6 m
(diameter of cyclones)
vertical length scale: lo4 m
(height of the troposphere)
time scale: 10’s
horizontal pressureg radient scale: 10m3m s-‘.
From the above estimateso ne can deducet he order of magnitude of the individual terms appearing in the
equations; this in turn may lead to the conclusion that certain terms can be neglectedw hen being small compared
with other terms in the same equation. One obtains the following expressionst ogether with an order of
magnitude estimation.
Land&-Biirnstein Fischer
New Series V/4a
1.2 Spherical coordinates 5
db ab
IS called the total time derivative, i.e. the temporal change of b following the motion; - is the local time
x at
derivative, i.e. the temporal change at a fixed point in space. - u3. V3b is defined as advection in meteorol-
ogy, V, ’ p,b is the divergenceo f the flux of b.
1.2 Spherical coordinates
The wind vector (like any other vector) may be written as
u,=iu+ju+kw,
where i, j, k are unit vectors pointing to the east, north, and upwards, respectively; u, v, w are the velocity
components.
d’ is denoted as zonal,
u=rcoscp3i
dv
u=r - is denoted as meridional, and
dt
dr
w= - as vertical velocity
dt
with 1 as longitude, rp as latitude and r as distance from the earth’sc entre.
The unit vectors are dependent on the coordinates I, cpa nd r in the following way:
a a a
ani = jsincp-kcoscp, -i =-i=O
acp ar
(2.1)
ik=isintp, &k= j; ;k=O.
1.2.1 The equations in spherical coordinates and the order of magnitude
of individual terms
For large-scalem otions (also called synoptic scalem otions) one presumesf rom observationsi n midlatitudes
the following characteristic orders of magnitude:
horizontal wind (u, u): 10m s- ’
vertical wind (w): lo-‘rns-’
horizontal length scale: lo6 m
(diameter of cyclones)
vertical length scale: lo4 m
(height of the troposphere)
time scale: 10’s
horizontal pressureg radient scale: 10m3m s-‘.
From the above estimateso ne can deducet he order of magnitude of the individual terms appearing in the
equations; this in turn may lead to the conclusion that certain terms can be neglectedw hen being small compared
with other terms in the same equation. One obtains the following expressionst ogether with an order of
magnitude estimation.
Land&-Biirnstein Fischer
New Series V/4a
6 1.2 Spherical coordinates
Equation of motion according to (1.2):
1 ap
~- + (2.2)
pcoscl, ai Fj,
Magnitude (in N. kg- ‘):
1O-4 10-3 10-6 10-e 10-s 1O-3 10-6
UN u2
+r +Ttan,=-ly +F,
er aq
du u2+ u 2 1a p
- g +F,
- -2Rcosq7~ I1 (2.4)
dt r = ear
Magnitude (in N. kg- ‘):
lo-’ 10-3 10-s 10’ 10’ 10-6
Continuity equation according to (1.3a):
(2.5)
Magnitude (in SK’):
10-6 10-s 10-s 1o-6
Thermodynamic equation (heat equation) according to (1.6):
c dr-!!E=, (2.6)
’ dt e dt
Magnitude (in W. kg- ‘):
10-l 10-l 10-2
Gas law according to (1.4):
ple=RT (2.7)
Magnitude (in J. kg- I):
10s 10s
In caset hat condensation of moisture takes place another equation for the conservation of moisture has to be
added (see 1.3b)):
i a SW
+ ;iar2w= -, (2.8)
en
where Q,,.i s the density of water vapor.
S,,. represents the sources (evaporation) and sinks (condensation) of water vapor (mass of water vapor per
volume and time).
1.2.1.1 Angular momentum
The zonal component of the equation of motion( 1.2) can also be expressedb y
1a p
rcosq(u+Rrcosq))= --- +rcosqFI [N.m.kg-‘1, (2.9)
e ai.
where r cos qo(u+ Rr cos p) is the component of the absolute angular momentum parallel to the axis of the earth.
frequently denoted as zonal angular momentum M, (see also 1.5.3 and (2.20)).
Fischer
Land&-BBrmtein
NW Sericr V,4a
1.2 Spherical coordinates 7
1.2.1.2 The vorticity equation
The vorticity equation, i.e. the rate of change of the vertical component of curlu, (see( 1.10)) has the form
~(i+f)=-(~+~)v.~+k.v~xv~-k. ( vwx (g+$)) fk.VXF (2.10)
Magnitude( in s- “):
10-10 10-10 10-11 lo-” 10-12
or
ac
---VS. v,(ifI)wcurlv,+iV,pxk+F,xk (2.11)
at ( >
with the relative vorticity [ =curlu, . k and f=2JZ. k=252 sincp as the vorticity of the solid earth (Coriolis
parameter).
Note that the operator V and the velocity vector II refer to the horizontal directions only (see 1.2.1.3a nd
1.2.2.1).
1.2.1.3 Vector operations
i ab j 3b
V,b= ~- tGG+k;=Vb+k;
rcoscp an
B = iB, + jB, + kB,
1.2.2 The hydrostatic system of equations
The order of magnitude estimation in (2.1) reveals that in the vertical component of the equation of motion
the pressure gradient force and the gravity force are by far dominating and practically balance each other. This
hydrostatic balance is always assumed to be valid in large-scale dynamics. Moreover, in the horizontal
components, (2.2) and (2.3), those terms containing the vertical velocity w, i.e. uw/r and vw/r, can be deleted,
becausew is about 2 orders of magnitude smaller than the horizontal velocity; for the same reason 2w/r in (2.5)
is neglected. Furthermore, the earth radius r=a+z m8y be replaced by the mean radius a, whenever it is
a a dr dz
undifferentiated, else we set - = -; w= - = -, where z is the height above sea surface.
ar aZ dt dt
The above approximations are due to the fact that the part of the atmosphere of interest is confined within a
“thin skin” around the earth. For this reason it is also appropriate to distinguish between horizontal motion
representedb y u and vertical motion w and to uset he “horizontal” operator V and the vertical differentiation a/az
separately.
The following physically consistent system of equations is obtained:
(2.12)
(2.13)
Landolt-Biirnstein Fischer
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8 1.2 Spherical coordinates
3r in vector form
de
d;+/kxu=-~vp+f [m.s-2=N.kg-1], (2.14)
where f= 2Rsinq is the Coriolis parameter.
The equations of motion now consist of only the horizontal components; the vertical component degenerates
to the hydrostatic equation
(2.15)
The continuity, heat and water vapor equations are from (2.5) (2.6) and (2.8):
(2.16)
(. dr-!!!t!=, [m*.s-3=W.kg-‘] (2.17)
’ dt P dt
ah, [kg.m-3.s-1]. (2.18)
ar +V~QH.u+ he‘w -&
The equation of state is given by (2.7).
The equations in this form are frequently called “primitive equations”, in order to distinguish them from the
modified equations in 1.4.2.3...1.4.2.5.
Upon scalar multiplication of (2.14) with u, the kinetic energy equation is obtained:
dK 1
-u.vp-D=-t.,.V,p-S,I~-D CW.kg-‘I, (2.19)
x=-&J
which has the same form as (1.9) except that K =i(zd’+ u2) and -D = II. F contain merely the horizontal wind.
The rate of change of “zonal” angular momentum, i.e. the momentum about the earth’sa xis, becomes
$f,= ~(ncosy(o+Qflcosy))= - ‘CJ! +acosy.F, [N.m.kg-‘1, (2.20)
e ai.
which is equivalent to (2.9).
By “cross-differentiation” of (3.1) and (3.2) or directly from (2.10) the vorticity equation follows:
~(~+/)=-(i+~)v.u+k.vpxv~-k.v\,~x E+k.VxF cs-2l> (2.21)
‘
which is nearly identical to (2.10).
1.2.2.1 Vector operations and Helmholtz theorem
Total time derivative:
db i3b
+nVb+\$
dr=at ‘
db a& a
pz = at +V.puh+ Fe\rb.
L
Horizontal gradient:
Horizontal divergence:
Fischer Landok-BBmclein
NCN Sericc v 4n
1.2 Spherical coordinates 9
Horizontal wind divergence:
1 au
V.v=- - + -a(VCOSO))
acosq ( %I &p >
Vorticity:
Jacobi Operator:
Laplace Operator:
v.Vb=V=b=
&(~(COS”~) +&&ii.
Helmholtz theorem
Since any vector can be decomposed into its rotational (nondivergent) and divergent part (Helmholtz
theorem), this results for the horizontal windvector into
u=kxVy,+VX=u,+u,.
rp is the streamfunction, uyrt he rotational part of the wind, x is the potential function and uXt he divergent part of
the wind. Then we have:
v.u=v.u,=v2~; {=kVxu,=V%p.
It can be deduced from (2.5) that the magnitude of V. u is lO-(j s-l, whereas that for [ is lo-’ SK’;
consequently also uyri s about one order of magnitude larger than uX.
1.2.2.2 Pressure tendency and Richardson equation
If: in (2.16)i s replaced by - 15 2from (2.15) and that equation is integrated from some level z to the top
g at aZ
of the atmosphere zT, then the pressure tendency equation is obtained:
ap ap
(-) -g3V.eudz+gew-(gew),. (2.22)
at= at, z
With the aid of (2.7), (2.16), (2.17), (2.22) one can derive Richardson’s equation:
(2.23)
with
J=u.Vp-gTV+udz; ws= w(zs).
z
With appropriate boundary conditions at the top z = zT and bottom z = zs of the atmosphere, (see 1.2.2.6):
ap
z -gew=o z=zT
(2.24)
w=u.Vh, z=zs,
where Vhs is the slope of the bottom topography, w is determined from the distribution of wind, pressure,
temperature and heating rate at a certain time; becausen o time derivative enters (2.23),t his is called a diagnostic
equation.
Land&-BBmstein Fischer
New Series V/4a
