Fundamentals Of Geophysical Fluid Dynamics Part 4 pptx

4.1 Rotating Shallow-Water Equations 117The Shallow-Water Equations can therefore be interpreted literallyas a model for barotropic motions in the ocean including effects of itsfree surf

Fig 4.2 Oceanic internal gravity waves on the near-surface pycnocline, asmeasured by a satellite’s Synthetic Aperture Radar reflection from the asso-ciated disturbances of the sea surface The waves are generated by tidal flowthrough the Straits of Gibraltar (NASA)

internal-gravity, inertial, and Rossby wave oscillations In this chapter

a more extensive examination is made for the latter three wave typesplus some others This is done using a dynamical system that is moregeneral than 2D fluid dynamics, because it includes a non-trivial in-fluence of stable buoyancy stratification, but it is less general than 3Dfluid dynamics The system is called the Shallow-Water Equations In

a strict sense, the Shallow-Water Equations represent the flow in a fluidlayer with uniform density, ρ0, when the horizontal velocity is constantwith depth (Fig 4.3) This is most plausible for flow structures whosehorizontal scale, L, is much greater than the mean layer depth, H, i.e.,H/L  1 Recall from Sec 2.3.4 that this relation is the same as-sumption that justifies the hydrostatic balance approximation, which isone of the ingredients in deriving the Shallow-Water Equations It isalso correct to say that the Shallow-Water Equations are a form of thehydrostatic Primitive Equations (Sec 2.3.5) limited to a single degree

of freedom in the vertical flow structure

4.1 Rotating Shallow-Water Equations 117The Shallow-Water Equations can therefore be interpreted literally

as a model for barotropic motions in the ocean including effects of itsfree surface It is also representative of barotropic motions in the atmo-spheric troposphere, although less obviously so because its upper freesurface, the tropopause, may more readily influence and, in response,

be influenced by the flows above it whose density is closer to the sphere’s than is true for air above water The Shallow-Water Equationsmimic baroclinic motions, in a restricted sense explained below, withonly a single degree of freedom in their vertical structure (hence theyare not fully baroclinic because ˆz· ∇∇p × ∇∇ ∇ρ = 0; Sec 3.1.1) Never-∇theless, in GFD there is a long history of accepting the Shallow-WaterEquations as a relevant analog dynamical system for some baroclinicprocesses This view rests on the experience that Shallow-Water Equa-tions solutions have useful qualitative similarities with some solutionsfor 3D stably stratified fluid dynamics in, say, the Boussinesq or Primi-tive Equations The obvious advantage of the Shallow-Water Equations,compared to 3D equations, is their 2D spatial dependence, hence theirgreater mathematical and computational simplicity

tropo-4.1 Rotating Shallow-Water Equations

The fluid layer thickness is expressed in terms of the mean layer depth,

H, upper free surface displacement, η(x, y, t), and topographic elevation

of the solid bottom surface, B(x, y):

Obviously, h > 0 is a necessary condition for Shallow-Water Equations

to have a meaningful solution The kinematic boundary conditions (Sec.2.1.1) at the layer’s top and bottom surfaces are

∂z(u, v) = 0 by assumption, the incompressible continuity relation plies that w is a linear function of z Fitting this form to (4.2) yields

im-w =

z

− Bh

Dη

Dt +

h + B

− zh



u· ∇∇B ∇ (4.3)

∂w

∂z =

1h

Shallow-4.1 Rotating Shallow-Water Equations 119The free-surface boundary condition on pressure (Sec 2.2.3) is p = p∗,

a constant; this is equivalent to saying that any fluid motion abovethe layer under consideration is negligible in its conservative dynamicaleffects on this layer (n.b., a possible non-conservative effect, also beingneglected here, is a surface viscous stress) Integrate the hydrostaticrelation downward from the surface, assuming uniform density, to obtainthe following:

∂p

=⇒ p(x, y, z, t) = p∗+

Z H+η z

gρ0dz0

=⇒ p = p∗+ gρ0(H + η− z) (4.7)

In the horizontal momentum equations the only aspect of p that matters

is its horizontal gradient From (4.7),

1

ρ0∇p = g∇∇ ∇∇η ;hence,

Du

Dt + f ˆz× u = −g∇∇η + F ∇ (4.8)The equations (4.1), (4.5), and (4.8) comprise the Shallow-Water Equa-tions and are a closed partial differential equation system for u, h, andη

An alternative conceptual basis for the Shallow-Water Equations isthe configuration sketched in Fig 4.4 It is for a fluid layer beneath aflat, solid, top boundary and with a deformable lower boundary separat-ing the active fluid layer above from an inert layer below For example,this is an idealization of the oceanic pycnocline (often called the ther-mocline), a region of strongly stable density stratification beneath theweakly stratified upper ocean region, which contains, in particular, theoften well mixed surface boundary layer (cf., Chap 6), and above thethick, weakly stratified abyssal ocean (Fig 2.7) Accompanying approxi-mations in this conception are a rigid lid (Sec 2.2.3) and negligibly weakabyssal flow at greater depths Again integrate the hydrostatic relationdown from the upper surface, where p = pu(x, y, t) at z = 0, throughthe active layer, across its lower interface at z =−(H + b) into the inertlower layer, to obtain the following:

p = pu− gρ0z , −(H + b) ≤ z ≤ 0

h have the same meaning as in Fig 4.3 Here pu is the pressure at the lid;

− b is elevation anomaly of the interface; ρl is the density of the lower layer;and g0= g(ρl− ρ0)/ρ0 is the reduced gravity The mean positions of the topand bottom are z = 0 and z =−H, respectively

p = pi = gρ0(H + b) + pu , z =−(H + b)

p = pi− gρl(H + b + z) , z≤ −(H + b) (4.9)(using the symbols defined in Fig 4.4) For the lower layer (i.e., z ≤

−(H + b) ) to be inert, ∇∇p must be zero for a consistent force balance∇there Hence,

4.1 Rotating Shallow-Water Equations 121The Shallow-Water Equations corresponding to Fig 4.4 are isomor-phic to those for the configuration in Fig 4.3 with the following identi-fications:

(b, g0, 0) ←→ (η, g, B) , (4.13)i.e., for the special case of the bottom being flat in Fig 4.3 In thefollowing, for specificity, the Shallow-Water Equations notation usedwill be the same as in Fig 4.3

4.1.1 Integral and Parcel Invariants

Consider some of the conservative integral invariants for the Water Equations with F = 0

Shallow-The total mass of the uniform-density, shallow-water fluid, ρ0M , isrelated to the layer thickness by

through the side boundaries, this expression can be manipulated to rive

The potential energy in (4.17) can be related to its more fundamentaldefinition for a Boussinesq fluid (2.19),

P E = 1

ρo

Z Z Z

dx dy dz ρgz (4.18)For a shallow water fluid with constant ρ = ρo, the vertical integrationcan be performed explicitly to yield

H+η B

2

Z Z

dx dy [H2+ 2Hη + η2− B2] (4.19)Since both H and B are independent of time andR R

dx dy η = 0 by thedefintion of H after (4.15),

KE The difference between P E and AP E is called unavailable potentialenergy, and it does not change with time for adiabatic dynamics Sinceusually H |η|, the unavailable part of the P E in (4.19) is much largerthan the AP E, and this magnitude discrepancy is potentially confusing

in interpreting the energetics associated with the fluid motion (i.e., theKE) This concept can be generalized to 3D fluids, and it is the usualway that the energy balances of the atmospheric and oceanic generalcirculations are expressed

4.1 Rotating Shallow-Water Equations 123

in the absolute vorticity, f (y2) + ζ2> f (y1) + ζ1> 0

There is another class of invariants associated with the potential ticity, q (cf., Sec 3.1.2) The dynamical equation for q is obtained bytaking the curl of (4.8) (as in Sec 3.1.2):

In the Shallow-Water Equations, in addition to the relative and tary vorticity components present in 2D potential vorticity (ζ and f (y),

plane-respectively), q now also contains the effects of vortex stretching Thelatter can be understood in terms of the Lagrangian conservation of cir-culation, as in Kelvin’s Circulation Theorem (Sec 3.1.1) For a materialparcel with the shape of an infinitesimal cylinder (Fig 4.5), the localvalue of absolute vorticity, f + ζ, changes with the cylinder’s thickness,

h, while preserving the cylinder’s volume element, h dArea, so that theratio of f +ζ and h (i.e., the potential vorticity, q) is conserved followingthe flow For example, stretching the cylinder (h increasing and dAreadecreasing) causes an increase in the absolute vorticity (f + ζ increas-ing) This would occur for a parcel that moves over a bottom depressionand thereby develops a more cyclonic circulation as long as its surfaceelevation, η, does not decrease as much as B does

The conservative integral invariants for potential vorticity are derived

by the following operation on (4.24) and (4.5):

Z Z

dx dy∇ · (Au) =∇

Z

ds Au· ˆn = 0 ,for A an arbitrary scalar, if u· ˆn = 0 on the boundary (i.e., thekinematic boundary condition of zero normal flow at a solid boundary),the result is

ddt

Z Z

This is the identical result as for 2D flows (3.29), so again it is truethat integral functionals of q are preserved under conservative evolu-tion This is because the fluid motion can only rearrange the locations

of the parcels with their associated q values by (4.24), but it cannotchange their q values The same rearrangement principle and integralinvariants are true for a passive scalar field (assuming it has a uniformvertical distribution for consistency with the Shallow-Water Equations),ignoring any effects from horizontal diffusion or side-boundary flux Theparticular invariant for n = 2 is called potential enstrophy, analogous toenstrophy as the integral of vorticity squared (Sec 3.7)

4.2 Linear Wave Solutions 1254.2 Linear Wave Solutions

Now consider the normal-mode wave solutions for the Shallow-WaterEquations with f = f0, B = 0, F = 0, and an unbounded domain Theseare solutions of the dynamical equations linearized about a state of restwith u = η = 0, so they are appropriate dynamical approximations forsmall-amplitude flows The linear Shallow-Water Equations from (4.5)and (4.8) are

= − H(∂tt+ f2)(∂xu + ∂yv) , (4.27)then substitute the x- and y-derivatives of the first two relations intothe last relation,

or for η06= 0, divide by −iη0 to obtain

ω(ω2− [f2+ c2k2]) = 0 (4.31)The quantity

c = p

is a gravity wave speed (Secs 4.2.2 and 4.5) Equation (4.31) is calledthe dispersion relation for the linear Shallow-Water Equations (cf., thedispersion relation for a Rossby wave; Sec 3.1.2) It has the genericfunctional form for waves, ω = ω(k) Here the dispersion relation is acubic equation for the eigenvalue (or eigenfrequency) ω; hence there arethree different wave eigenmodes for each k

Wave Propagation: The dispersion relation determines the tion behavior for waves Any quantity with an exponential space-timedependence as in (4.29) is spatially uniform in the direction perpendic-ular to k at any instant, and its spatial pattern propagates parallel to k

propaga-at the phase velocity defined by

cp = ω

However, the pattern shape is not necessarily preserved during an tended propagation interval (i.e., over many wavelengths, λ = 2π/|k|,and/or many wave periods, P = 2π/|ω|) If the spatial pattern is asuperposition of many different component wavenumbers (e.g., as in aFourier transform; Sec 3.7), and if the different wavenumber compo-nents propagate at different speeds, then their resulting superpositionwill yield a temporally changing shape This process of wavenumberseparation by propagation is called wave dispersion If the pattern has adominant wavenumber component, k∗, and its amplitude (i.e., the coef-ficient of the exponential function in (4.29)) is spatially localized withinsome region that is large compared to λ∗= 2π/k∗, then the region thathas a significant wave amplitude will propagate with the group velocitydefined by

ex-cg = ∂ω

∂k

Thus, one can say that the wave energy propagates with cg, not cp If

cp6= cg, the pattern shape will evolve within this region through sion, but if these two wave velocities are equal then the pattern shapewill be preserved with propagation Waves whose dispersion relation

disper-4.2 Linear Wave Solutions 127implies that cp = cg are called non-dispersive There is an extensivescientific literature on the many types of waves that occur in differentmedia; e.g., Lighthill (1978) and Pedlosky (2003) are relevant booksabout waves in GFD.

4.2.1 Geostrophic ModeThe first eigenenvalue in (4.31) is

4.2.2 Inertia-Gravity WavesThe other two eigenfrequency solutions for (4.31) have ω6= 0:

ω2 = [f2+ c2k2]

=⇒ ω = ± [f2+ c2K2]1/2, K =|k| (4.37)First take the long-wave limit (k→ 0):

direction of the wavenumber vector, ˆek = k/K Waves in the limit(4.39) are non-dispersive Since any initial condition can be represented

as a superposition of k components by a Fourier transform (Sec 3.7),

it will preserve its shape during propagation In contrast, waves nearthe inertial limit (4.38) are highly dispersive and do not preserve theirshape

For the linear Shallow-Water Equations, the Br¨unt-V¨ais¨all¨a frequency(Sec 2.3.3) is evaluated as

in (4.41); this, rather than KH→ ∞ and the resulting (4.39), is about

as far as the short-wave limit should be taken for the Shallow-WaterEquations due to the derivational assumption of hydrostatic balanceand thinness, H/L 1 (Sec 4.1) Recall from Sec 2.3.3 that ω = ± N

is the frequency for an internal gravity oscillation in a stably stratified3D fluid (In fact, this is the largest internal gravity wave frequency in

a 3D Boussinesq Equations normal-mode solution; n.b., exercise #6 forthis chapter.) The limit (4.39) is identified as the gravity-wave modefor the Shallow-Water Equations It can be viewed alternatively as anexternal or a surface gravity wave for a water layer beneath a vacuum

or an air layer (Fig 4.3), or as an internal gravity wave on an interfacewith the appropriately reduced gravity, g0, and buoyancy frequency, N(Fig 4.4)

It is typically true that ”deep” gravity waves with a relatively largevertical scale, comparable to the depth of the pycnocline or tropopause,have a faster phase speed, c, than the parcel velocity, V Their ratio iscalled the Froude number,

4.2 Linear Wave Solutions 129atmosphere andO(1) m s−1in the ocean For the V values characteriz-ing large-scale flows (Sec 2.4.2), the corresponding Froude numbers are

F r ∼ Ro in both media Thus, these gravity waves are rapidly agating in comparison to advective parcel movements, but also recallthat sound waves are even faster than gravity waves, with M  F r(Sec 2.2.2)

prop-Based on the short- and long-wave limits (4.38)-(4.39), the second set

of modes (4.37) are called inertia-gravity waves, or, in the terminology

of Pedlosky (Sec 3.9, 1987), Poincar´e waves Note that these modesare horizontally isotropic because their frequency and phase speed,|cp|,are independent of the propagation direction, ˆek, since (4.37) dependsonly on the wavenumber magnitude, K, rather than k itself

For inertia-gravity waves the approximate boundary between the dominantly inertial and gravity wave behaviors occurs for KR = 1,where

|f| =

√gH

in the configuration in Fig 4.4, as well as in 3D stratified fluids (Chap.5) Internal deformation radii are much smaller than external ones be-cause g0  g; typical values are several 100s km in the troposphere andseveral 10s km in the ocean

For the inertia-gravity modes, the modal amplitude for vorticity is

q−Hf = f + ζ

H + η −Hf ≈ Hζ −Hf η2 (4.45)

prop-Based on the short- and long-wave limits (4. 38)- (4. 39), the second set

of modes (4. 37) are called inertia-gravity waves, or, in the terminology

of Pedlosky (Sec 3.9, 1987), Poincar´e... booksabout waves in GFD.

4. 2.1 Geostrophic ModeThe first eigenenvalue in (4. 31) is

4. 2.2 Inertia-Gravity WavesThe other two eigenfrequency solutions for (4. 31) have ω6= 0:

ω2... resulting (4. 39), is about

as far as the short-wave limit should be taken for the Shallow-WaterEquations due to the derivational assumption of hydrostatic balanceand thinness, H/L (Sec 4. 1)