The Momentum Equation

Hydrodynamic Equations of the Sea

3.11 The Momentum Equation

Hydrodynamic Equations of the Sea 105

Here the diffsivity of salt in seawater, ^K,, depends on the thermodynamic state of the fluid, i.e., temperature, pressure, etc. For isotropic molecular diffusion alone, ^K, = 1.5 x m2 s - ' at T = 25"C, and at normal oceanic salinities. As with eddy viscosity, the eddy diffusion of salt is much larger than molecular diffusion, of course, but less is known about numeri- cal values for salinity eddy diffusion than for momentum. However, esti- mates from the Mediterranean salt tongue of Figs. 2.24 and 2.25 give values of

K , ~ = 3 x 103 m 2 s - ' and

K,, = 5 x m 2 s - '

,

(3.82) which are somewhat smaller than the momentum diffusivities. The source/

sink terms represent either surface or volumetric rates for evaporation, E, minus precipitation, P, and freezing, F, minus thawing, 8 , minus R , the in- put of fresh water from riverine sources. This equation, together with anal- ogous ones for momentum and heat, can be used to describe thermohaline circulations, salt-fingering and other density-driven motions.

with all of the major terms displayed explicitly except for the centripetal con- tribution.

To recapitulate Eq. 3.83 term by term: The first is the acceleration at a fixed point, the so-called inertial term; the second is the advective nonlinear term giving the velocity change along a trajectory; the third is the Coriolis acceleration; the fourth is the combined gravity and centripetal accelerations;

the fifth is the pressure force per unit mass; and the sixth is the combined eddy and viscous decelerations. Collectively they describe most of the known volume forces acting on a unit mass of ocean. Other forces may also be in- cluded as boundary forcing terms, or implicit thermodynamic effects, and these will be introduced as the case requires. This form of the momentum equation with the molecular viscosity included but without the eddy viscosi- ty was originally due t o Navier and Stokes and is known by those names even today.

It is instructive to write the momentum equation in component form and we will do so in Cartesian coordinates, for use in the tangent-plane geome- try. In spherical coordinates, which must be used in planetary-scale numeri- cal models, the equations are much more complicated. In Cartesian coor- dinates, we have defined u, u, and w to be the x, y , and z components of current velocity, respectively; in spherical coordinates, however, they become the east

(9,

or zonal), north (A or meridional), and vertical (r, or up). Note that we have shifted from using the colatitude, 8, t o the latitude, A, so that the direction of “north” and the associated unit vectors in both the spheri- cal and tangent-plane systems have the same sense, i.e., toward the pole, and so that the tangent-plane system may be a slightly better approximation to the spherical system. This means that in order to maintain a right-handed coordinate system, the ordered triads of vectors in the two systems have differ- ent sequences than the usual one of r, 8, and

9.

This system is then one of geographical latitude and longitude (A,$), rather than the usual mathemati- cal spherical coordinates (8,q5), although only the transformation 8 = n/2

- A has been made. It should be noted that geographical coordinates (as distinct from spherical) use 4 for the latitude and X for the longitude. In spite of their widespread use in geophysics, we have chosen not to write the salient equations in geographical coordinates because of the confusion that might result. The transformations from spherical to geographical coordinates are, respectively,

and

Hydrodynamic Equations of the Sea

Momentum Equations in Cartesian Coordinates (1) x component:

a@

1 ap 1

ax ax

+

^- v * A . v u .

_ _ _ - _

(2) y component:

aU aU aU au

a t ax aY az

-

+

^{u -}

+

^{u -}

+

w ^{- =} -2uClsinA

(3)

z

component:

where the viscous operator is (from Eqs. 3.57 and 3.60)

+

- ( A ” $ )

a +

^{p v 2 .}

az

107

(3.84)

(3.85)

(3.86)

(3.87)

Momentum Equations in Spherical Coordinates

The momentum equations in spherical coordinates are much more com- plicated than in x,y,z coordinates, and even more so if the oblate spheroidal coordinates appropriate to Fig. 3.2 are used. An additional complication of

the latter system arises because gravity is such a dominant force in the equa- tions that even a very small but erroneous component of - V $ in the local horizontal direction, arising from an incorrect specification of a coordinate system, would constitute an important forcing term. For this reason an ide- al if complex coordinate system could be defined by equipotential surfaces such as shown in Fig. 3.7, with z locally perpendicular to that surface. A.

E. Gill discusses this as applied to an oblate spheroid system that approxi- mates the geoid to within 0.2%. Here we shall simply assume that the sys- tem is spherical, then write appropriate expressions for the various operators in Eq. 3.83, and apply those t o the momentum equation for this case. The other fluid equations will follow from the application of the operators listed below.

(1) Convective derivative:

(3.88)

u

ay ay

^{a Y}

+ - - +

^{- -}

+

^{w - .}

- a7

Dt a t r cos A ad r a A ar (2) Gradient operator:

(3.89)

(3) Divergence operator:

(3.90)

a

i a

V . F = ^~

[:

^-

⁺

~ ( F , c o s A ) r cos A

(4) Viscosity operator (assuming A constant and isotropic):

1 r2 cos2 A

a

(3.91)

Hydrodynamic Equations of the Sea

2 sin A a u

+

2w 2 au

2

a

r2 cos A a A

-~ - ( v cos A)] ,

1

aZy + - -

¹

a

(cos A:) r2 cos A

aa

r2 cos2 A

Zj?

Ay =

i a

+

^{- -} ( t - 2 ; ) .

r2 ar

109

(3.92)

(3.93)

(3.94)

( 5 ) 4 component:

~ Du = (2Q

+

r&A)(v sin A - wcos A ) Dt

(3.95)

+ A 9) +

^- 1 (AV2U)@, , r c o s A

(6

^p a4 P

- - 1

( 6 ) A component:

VW (2Q

+ -)

^{U u sin A}

- _ _

^-

DV -

Dt r r cos A

-

1 P

(7) r component:

Dw

v 2 U

~ _Dt = - _r

+

(2Q

+

_r

-)

_{cos A} ^{u cos A}

-

(; ⁺ ; ^{$) + p}

1 ^(AV2U),

^.

(3.96)

(3.97)

Because the depth of the ocean is such a small fraction of the radius of the earth (0.08%), a useful approximation is to assume that r in these equa- tions is simply R e , and that the radial derivatives can be replaced by l)/&,.

The full spherical coordinate system must be used, rather than the beta plane approximation, whenever the north-south scale of the motion is large enough for variations in the numerical value of

6

to be important; another instance is when the azimuthal connectedness of the motion plays a role, as is the case for the Antarctic Circumpolar Current.