THE NAVIER-STOKES EQUATION

θ

~

2~

not add to zero, the energy balance B is nonzero. As

a result, this energy imbalance must reflect phase

t

xj

xj xj

changes (freezing or melting) or warming or cooling

I

II

III

(storage or release of heat) of the ice within the floe.

As a boundary-layer meteorologist, I could justifi-

Here, term I is the time rate of change of the instanta-

ably study any but the C term on the right-hand side of

neous temperature. Term II quantifies advective effects

eq 3a. But since my main interest is turbulence in the

on the temperature. Term III contains the molecular ef-

ABL, I will confine my attention here to Hs and HL.

fects on the temperature, where D is the thermal diffu-

These are the only two turbulence terms in eq 3. As I

v

sivity of heat in the fluid.

said for τ , the desire to know Hs and HL is the basis for

In the atmosphere, the instantaneous specific humid-

most of what I will write here.

~

ity q is usually another conservative scalar. We can write

an equation for it with exactly the same form as eq 7

THE NAVIER-STOKES EQUATION

All geophysical fluid dynamics starts with the

t

xj

xj xj

Navier-Stokes equation (e.g., Busch 1973, Businger

1982)

where Dw is the molecular diffusivity of water vapor in

gδi3 2Ωε ijk ηj uk + ν

Equation 4, especially, is too complex to treat as it

stands; we need to simplify it. Our first simplification is

to recognize that each instantaneous quantity in eq 4, 7

(4)

and 8 can be decomposed into an average and a turbu-

In my notation, a wavy overbar indicates an instanta-

lent fluctuation about that average. That is,

neous value; ui , for example, is the ith component of

~

ui = Ui + ui

the instantaneous fluid velocity vector. Thus, eq 4 shows

~

(9a)

the processes that affect this component.

~

θ=Θ+θ

(9b)

In eq 4, term I is the time rate of change (the accel-

eration) of the ith component. Term II really contains

q =Q+q

~

(9c)

three terms since the repeated j index indicates a sum.

It shows the advective effects on the ith velocity com-

ρ = ρ0 + ρ

~

ponent. In term III, ρ is fluid density and p is the pres-

(9d)

~

sure. Term III, thus, shows the effects of a pressure gra-

p = P + p.

~

(9e)

dient on the fluid--fluid tends to accelerate down the

gradient. In term IV, g is the acceleration of gravity,

In eq 9, a capital letter indicates an averaged quantity,

and δij is the Kronecker delta, where δij = 1 if i = j, and

except in eq 9d where ρ0 indicates the average air den-

δij = 0 if i ≠ j. Term IV therefore represents the gravita-

sity. A lower case letter without the wavy overbar is a

~

tional force and affects only the u3 component of the

zero-mean turbulent fluctuation about that average.

velocity (the vertical component). Term V shows the

Let me demonstrate how to manipulate these decom-

Coriolis effects on the fluid motion. Here Ω is the Earth's

posed quantities. Let a straight overbar indicate averag-

rotation rate, 2π radians per day (7.27 105 s1), and ηj

~

ing. Then, with ui for example

is the unit vector of the Earth's rotational axis

ui ≡ Ui

v

(10)

~

η = (0, cos λ, sin λ)

(5)

since the average of ui is zero by definition ( ui ≡ 0 ). Simi-

~

where λ is the latitude. Notice, the quantity f in eq 1 is

larly

f = 2Ω sinλ .

uiuj = (Ui + ui ) (U j + uj )

(6)

~~

(11a)

= UiU j + uiU j + ujUi + uiuj

Finally, term VI quantifies viscous effects on the flow,

(11b)

where ν is the kinematic viscosity of the fluid.

= UiU j + uiuj

We can write an equation analogous to eq 4 for the

(11c)

conservation of any conservative scalar--potential tem-

~

perature θ , for example (Busch 1973, Businger 1982)

since Ui and Uj are already averaged quantities.

2