Quasi-Steady Motions

Waves and Tides

5.2 Quasi-Steady Motions

Even in the steady state, the effects of Coriolis force and stratification are dominant in geophysical fluids. We will now derive two equations that characterize the horizontal and vertical forces at work in the case of steady currents in a stratified ocean, the geostrophic equation and the Brunt- Vaisala frequency relationship.

In the steady state, time derivatives are zero, by definition. Additionally, we confine our attention for now to regions where the spatial variations of current are negligible, and will also neglect tidal forcing as well. Under these conditions, the horizontal component of the momentum equation, as was discussed in Section 2.8, takes on the simple form

which can be decomposed into x and y components using Eqs. 3.84 and 3.85:

1 aP

P ax

2Qv sin A = - - ,

and

1 aP -2Qu sin A = - - ,

P aY

respectively, or, with a slight rearrangement, aP - = pfv

ax and

These are the equations of geostrophic balance, which for the Northern Hemi- sphere state that the flow of a northward steady current, u , is subject to an eastward Coriolis force (Fig. 3.6) and is balanced by a westward horizontal pressure gradient; additionally, a steady eastward current, u (southward Cori- olis force), is balanced by a northward pressure gradient. The pressure gra-

Waves and Tides 163

dients are developed during the ocean's adjustment to equilibrium by way of an inclination or tilting of its constant density surfaces away from the equipotential surfaces, which sets up a hydraulic head (or elevation above the equipotential) of approximately the right amount necessary to balance the current against the Coriolis force. This is termed geostrophic flow, and will be explained more fully ahead. In the Southern Hemisphere, the Corio- lis force and the sense of the gradients are reversed, but the dynamic balances remain the same.

The vertical component of the momentum equation in the steady state yields the simple hydrostatic equation:

where g is now taken t o be the effective gravity as given by the sum of Eqs.

3.24 and 3.46. Since g =

-gk,

we have

This is immediately integrable if p ( z ) is known from the surface

( z

⁼ 0) down to the depth

z

⁼ - h :

where p a is the atmospheric pressure, as before. Thus the absolute pressure at depth - h can be calculated, knowing the vertical density distribution. To a first approximation, the density, which varies roughly between 1028 and 1050 kg m ^{- 3} under typical deeper oceanic conditions, can be assumed con- stant at, say, pr = 1036 kg m - 3 . Then the pressure is simply

Since g ⁼ 9.80 m s - * ,

p ( - h ) ^- p a = 1036 x 9.80 h

= 10153 h , (5.10)

where h is in meters a n d p - pu is in pascals. Because lo5 Pa = 1 bar, this gives rise to the rule of thumb that the oceanic pressure increases at approxi- mately 1 decibar per meter of depth. Oceanic pressures are therefore often quoted in dbar, which are numerically within 1 to 2% of the depth in meters.

I t should be noted that the approximation of constant density is not suffi- cient for calculating the horizontal pressure gradients needed in Eqs. 5.4 or 5 . 5 . There the full equation of state is required, with in situ values of tem- perature, salinity, and pressure being used in the calculation, as was discussed in Chapter 4.

In order to obtain a more tractable description of the variations of pres- sure and density, let us separate each of these quantities into vertically vary- ing, steady state components, p o ( z ) and p o (z), respectively, plus fluctuating, time-dependent perturbation components, p ' ( x , t ) and p ' ( x , r ) , respectively.

The perturbation components are assumed t o be small variations about the background equilibrium values of pressure and density. Thus we may write:

Now an important and useful variant of the momentum equations may be derived by substituting these into Eqs. 4.51 to 4.53. Neglecting the tidal poten- tial,

a,,

the terms in pressure and gravity become

v p ' - (5.14)

which, because of the hydrostatic equation (Eq. 5.7), becomes a relation- ship with only the primed quantities appearing in the pressure and gravity terms of the momentum equation:

Du

p - Dt = -2 p a x u - V p '

+

^{p ' g}

+

V - A - V u . ^(5.15)

Waves and Tides 165

Note that the total density, p o

+

^{p ‘ ,} has been retained in the acceleration and Coriolis terms. The quantity p ’ g / p is called reduced gravity, g ’ , and is typically g in the ocean because of the small size of density pertur- bations :

P - Po P’ - g = ^~

P P

g = g ’ . (5.16)

Its negative, ^- g ’ , is sometimes called the buoyancyforce, since a parcel of water of density near po

+

^{p ’} is buoyed up by its surroundings, so that the effective gravitational force acting on it is reduced to g ’ . Reduced gravity is of importance in baroclinicflows, where the velocity and density vary rapid- ly in the vertical, especially near the surface layers; such motions are typical- ly of much lower frequency and propagate more slowly than the barotropic flows, which are essentially uniform throughout the water column.

A further simplification to the momentum equations that is of considera- ble utility is called the Boussinesq approximation, in which the actual densi- t y multiplying the inertial term Du/Dt is replaced by a constant density, p c .

This retains the reduced gravity in the buoyancy forces but does not elimi- nate acoustic waves and compressibility and, as a bonus, simplifies certain analyses. It is especially useful in the shallow-water hydrostatic approxima- tion. We will use the Boussinesq approximation for much (but not all) work ahead.

Returning to the hydrostatic equation, the unperturbed pressure and den- sity may be substituted into Eq. 5.7 to obtain

(5.17) where co is the speed of sound. Equation 5.17 may be immediately integrat- ed from the surface, where p o = p s , t o depth

z:

g dz’

(5.18) For a constant speed of sound, co, this gives

where ^I, is negative below the surface of the sea. If c varies in the vertical,

it is natural then to define a local scale height, H,

( z ) ,

which is the height over which the density increases by e:

H, ( 2 ) = c 2 ( z ) / g

.

^(5.20)

Since H, is approximately 225 km in the ocean, many problems having small amplitudes of motion allow the constant density approximation.

Nevertheless, Eq. 5.15, in which we have not made the Boussinesq approxi- mation, has greater generality and will be used in a number of problems re- quiring it.