Interior mixing and phytoplankton survival in stratified environments - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 206

To establish a more fundamental physical model (Simpson, Crawford, et al., 1996 )
we start with the horizontal momentum Equations (3.9) which, in linearised form
without external forces, may be written as:

@ u 1 @ p

x þ @

t x
@

@ v 1 @ p

y þ @

t y
@

t ¼

;

t ¼

; : ð : Þ

fv fu 7 1

@ @ z @ @ z

The motion is forced by the tides through an oscillating pressure gradient which, in the
absence of horizontal density gradients, may be written in terms of the surface slopes as:

@ p

g @
@

@ p

g @
@

x ¼ x ; y ¼ y :

ð : Þ

7 2

@ @

The slopes are conveniently found as a sum of N harmonic constituents (see Section
2.5.1 ) of the surface slope induced by the tide; e.g. for the x component:

X

N

@
@ x ¼

A n cos ð! n t þ n g n Þ ð 7 : 3 Þ

n ¼ 1

where A n and g n are the amplitude and phase lag of the surface slope constituent n
with frequency o n . The phase a n is the lag of constituent n in the TGF.
Wind forcing enters at the surface boundary via the stress terms (t Wx , t Wy ) which
are related to the velocity shear by the eddy viscosity N z as in Equation (4.37). At the
surface (z ¼ 0), the stress is set equal to the applied wind stress so that:

N z @

u

N z @

v

t x ¼

z ¼

t Wx ; t y ¼

z ¼

t Wy : ð 7 : 4 Þ

@ @

At the bottom boundary (z ¼ h), the stresses must match the bottom stress in the

quadratic drag law

N z @

u

N z @

v

z ¼

k b Uu b ;

z ¼

k b Uv b ; ð 7 : 5 Þ

@ @

q

u b þ where U is the modulus of the bottom velocity (u b , v b ).

Density will change in response to changes in T and S which are governed by the
advection-diffusion Equation (4.39). In the absence of horizontal gradients and
assuming the mean vertical motion is zero, this reduces to:

¼ v b

@ð T ; S Þ

¼ @
@

K z @ð

T ; S Þ

ð 7 : 6 Þ

@ t z @ z

where K z is eddy diffusivity which we take to be the same for heat and salt.
Surface heat exchange consists of solar radiation input of Q s (1

A) and heat losses

Q u as in the TML model. At the surface boundary (z ¼ 0), the diffusive heat flux must

c p 0 K z @ T

equal the net heat input to the top layer, i.e. Q u . Also,

if we assume that precipitation and evaporation are in balance, then there is no
exchange of freshwater so that @ S

@ z ¼

0 : 55 Q s ð 1 A Þ

@ z ¼

0. Similarly at the bottom (z ¼ h) the fluxes of

heat and salt are negligible, so we can set @ T

@ z ¼ @ S @ z ¼

0.
The Equations (7.1) and (7.6) may be numerically integrated forward in time
for specified surface slope, wind stress and surface heat exchange providing we know