4.2.1.1 General Features of the Model
The RSW equations in the f -plane approximation with no dependence on the y-coordinates (i.e.∂y. . .≡0) are
∂tu+u∂xu− fv+g∂xh =0,
∂tv+u∂xv+ f u=0, (4.1)
∂th+∂x(uh)=0.
x v(x,t)
h(x,t) (x,t)
g Ω
u
Fig. 4.1 Schematic representation of the 1.5d RSW model
Here u, v are the across-front and the along-front components of the velocity, respectively, h is the total depth (no topographic effects will be considered in what follows), g is gravity (or reduced gravity – see below), f is the Coriolis parameter, which will be supposed constant (the f -plane approximation), unless the opposite is explicitly stated, and the subscripts denote the corresponding partial derivatives.
A sketch of the plane-parallel RSW configuration is presented in Fig. 4.1.
The model possesses two Lagrangian invariants: the generalized (geostrophic) momentum M =v+ f x and the potential vorticity (PV) Q =vxh+f:
(∂t +u∂x)M =0, (∂t +u∂x)Q=0, (4.2) which are related: Q = ∂xhM. Let us emphasize that the conservation of the geostrophic momentum is a consequence of 1.5 dimensionality of the problem. The straightforward linearization around the state of rest h = H0=constant gives the zero-frequency (slow) mode (the linearized PV) and the fast surface inertia - gravity waves with the dispersion law:
ω= ±(c02k2+ f2)12, (4.3) where c0 = √
g H0is the “sound speed”, i.e. the maximum phase speed of short inertia-gravity waves,ωis the frequency and k is the wavenumber.
The geostrophic equilibria are steady states:
fv=g∂xh. (4.4)
They are the exact solutions of the full nonlinear equations (4.1), which makes a dif- ference with respect to the full 2d RSW equations, where the geostrophic equilibria are not solutions, but are just slow (e.g. Reznik et al. [20]).
4.2.1.2 Lagrangian Approach to 1.5d RSW
In order to fully exploit the existence of a pair of Lagrangian invariants in the model, it is natural to introduce the Lagrangian coordinates X(x,t)of the fluid “parcels”
(in fact, fluid lines along the y-axis). They are given by the mapping x → X(x,t), where x is a fluid parcel position at t = 0 and X – its position at time t. Hence
X˙ ≡∂tX =u(X,t). The momentum equations in (4.1) become:
X¨ − fv+g∂h
∂X =0, (4.5)
∂t(v+ f X)=0, (4.6)
wherevis considered as a function of x and t. The mass conservation for each fluid element h(X,t)d X =hI(x)d x means that
h(X,t)=hI(x)∂x
∂X. (4.7)
This equation, obviously, is equivalent to the continuity equation in (4.1). Equation (4.6) immediately gives
v(x,t)+ f X(x,t)=vI(x)f x =M(x) . (4.8) By applying the chain differentiation rule to (4.7) and injecting the result into (4.5) we get a closed equation for X :
X¨ + f2X+ghI 1
(X)2 +ghI
2 1
(X)2
= f M, (4.9)
where prime denotes∂x. In terms of the deviations of fluid parcels from their initial positions X(x,t)=x+φ(x,t)(4.9) takes the form:
φ¨+ f2φ+ghI 1
(1+φ)2 +1 2ghI
1 (1+φ)2
= fvI. (4.10)
This single equation is equivalent to the whole system (4.1). It should be solved with initial conditionsφ(t =0)=0;φ(t˙ =0)=uI(x). Thus, the Cauchy (adjustment) problem is well and naturally posed for this equation.
It should be noted that 1.5d RSW in Lagrangian variables may be as well formulated in theβ-plane approximation, i.e. taking into account the dependence of the Coriolis parameter on latitude: f = f0+βy. For example, for purely zonal flows on the equatorialβ-plane ( f0≡0) we get
Y¨+βY u+g∂h
∂Y =0, (4.11)
∂t
u−βY2 2
=0,
h(Y,t)=hI(y)∂y
∂Y, (4.12)
and the closed equation for Y follows:
Y¨+βY
uI+βY2−y2 2
+ghI 1
(Y)2 +ghI
2 1
(Y)2
=0, (4.13)
to be solved with initial conditions Y(y,0)=y,Y˙(y,0)=vI(y).
4.2.1.3 The Slow Manifold
By additional change of variables x = x(a), the elevation profile in (4.5), (4.6), and (4.7) may be “straightened” to a uniform height H in order to have J = ∂∂Xa =
H
h(X,t). It is easy to see that ∂∂Xh = ∂∂Pa, where P = 2 Jg H2 is the so-called Lagrangian pressure variable. The Lagrangian equations of motion then take the form:
˙
u− fv+g H ∂
∂a 1
2 J2 =0, (4.14)
˙
v+ f u=0, (4.15)
J˙−∂au=0, (4.16)
and may be again reduced to a single equation:
J¨+ f2J+∂2P
∂a2 = f H Q, (4.17)
where Q – potential vorticity as a function of the a variable is Q(a)
= H1 ∂v
∂a+ f J
= H1
∂vI
∂a + f JI
.
The slow manifold is the stationary solution of (4.17) or (4.9). By re-introducing the X -variable and the dependent variableη= Hh we get
− g f
d2h(X)
d X2 +h(X)Q(X)= −f. (4.18)
Note that potential vorticity in terms of initial height and velocity fields reads Q(X(x))= f+h∂vII∂x . The following theorem may be proved by standard methods of
ordinary differential equations (Zeitlin et al. [25]): Equation (4.18) has a bounded and everywhere positive unique solution h(X)on R for positive Q(X)with compact support and constant asymptotics (frontal case).
It should be noted that positiveness of Q corresponds to the absence of the so- called inertial instability (see the next section). The latter is related to the presence of sub-inertial (i.e. ω < f ) frequencies in the spectrum of small excitations of the adjusted state. It may be, however, explicitly shown either in Eulerian variables (Zeitlin et al. [25]) or in Lagrangian variables (see below) that the spectrum of small perturbations over an adjusted front in 1.5d RSW is supra-inertial. Although we have no proof for non-positive distributions of Q, direct numerical simulations (Bouchut et al. [4]) indicate that a unique adjusted state is always achieved in this case too.
4.2.1.4 Relaxation Towards the Adjusted State
Once the existence of the adjusted state is established, the process of relaxation towards this state may be analysed. The first step in studying relaxation is lineariza- tion around the adjusted state:
u = ˜u, v=vs+ ˜v, J =Js+ ˜J
∂tu˜− fv˜−g H∂a(J˜/Js3)=0, (4.19)
∂tv˜+ f u=0, (4.20)
∂tJ˜−∂au=0, (4.21)
where the Lagrangian time derivative is denoted by∂t from now on. By using
fJ˜+∂av˜=0, (4.22)
it is easy to get a single equation forJ and/or for˜ v˜
∂t t2J˜+ f2J˜−g H∂aa2 (J/˜ Js3)=0, ∂t t2v˜+ f2v˜−g H∂a(˜va/Js3)=0. (4.23) Let us consider stationary solutions
J˜= ˆJ(a)e−iωt+c.c., v˜= ˆv(a)e−iωt+c.c.. (4.24) Then the stationary equations are
∂aa2 (g HsJ)ˆ +(ω2− f2)Jˆ=0, (4.25)
∂a(g Hs∂av)ˆ +(ω2− f2)vˆ=0, (4.26) where we denoted Hs = H/Js3. The equation for vˆ is self-adjoint and supra- inertiality ofωand, hence, the absence of trapped states follows trivially from (4.26) by multiplying byvˆ∗and integrating by parts:
ω2= f2+
da g Hs∂avˆ2
da|ˆv|2 ; ⇒ω2≥ f2. (4.27)
By using a new dependent variable ˆ
v= ψ
g Hs1/2
, (4.28)
we transform the stationary equation to a two-term canonical form d2ψ
da2 +
ω2− f2 g Hs −1
4
(Hs)a
Hs
2
−1 2
(Hs)a
Hs
a
ψ=0. (4.29)
Rewritten as
d2ψ
da2 +kψ2(a)ψ=0, (4.30)
this equation can be interpreted as that of a quantum mechanical oscillator with variable frequency kψ(a)(or as a Schrửdinger equation with a potential V and an energy E such that kψ2 =E−V(a)). It is clear that kψ2 can be negative forω > f and suitable Hs. This means that for certain intervals on the x-axis the wavenumber kψ may be imaginary and, hence, quasi-stationary states slowly tunneling out such zones may exist. Thus, the wave motions can be maintained for long times in such locations.
4.2.1.5 Wave Breaking
The direct simulations of the Lagrangian equations of motion indicate that singu- larities (shocks) may appear in the emitted inertia-gravity field. In the context of adjustment, shocks could provide an alternative sink of energy, whence the impor- tance to establish the criteria of wave breaking and shock formation. Shocks are of no surprise in gas dynamics, and the shallow-water equations are a particu- lar case of it. The only question, thus, is the role of rotation in this process. The Lagrangian approach, again, proves to be efficient (Zeitlin et al. [25]). The dimen- sionless Lagrangian equations of motion in a-variables introduced above are
∂tu+∂ap=v ,
∂tJ−∂au=0, (4.31)
wherevis not an independent variable and is to be found from∂av=Q(a)−J . We thus have a quasi-linear system
∂t
u J
+
0 −J−3
−1 0
∂a
u J
= v
0
. (4.32)
The eigenvalues of the matrix in the l.h.s. of (4.32) are μ± = ±J−32 and the corresponding left eigenvectors are
1,±J−32
. Hence, Riemann invariants are w±=u±2 J−12 and we have
∂tw±+μ±∂aw±=v . (4.33)
Expressions of original variables in terms ofw±are
u =1
2(w++w−) , J = 16
(w+−w−)2 >0, μ±= ±
w+−w− 4
3
. (4.34)
In terms of the derivatives of the Riemann invariants r±=∂aw±, we get
∂tr±+μ±∂ar±+∂μ±
∂w+r+r±+∂μ±
∂w−r−r±=∂av=Q(a)−J, (4.35) which may be rewritten using Lagrangian derivatives along the characteristicsdtd
± =
∂t+μ±∂aas
dr± dt± +∂μ±
∂w+r+r±+∂μ±
∂w−r−r±=Q(a)−J. (4.36) Wave breaking and shock formation correspond to r±→ ±∞in finite time.
In terms of new variables R±=eλr±, withλ= 1283 log|w+−w−|, (4.35) may be rewritten as
d R±
dt± = −e−λ∂μ±
∂w±R±2 +eλ(Q(a)−J) , (4.37) where ∂w∂μ±
± =643(w+−w−)2>0.
The qualitative analysis of these generalized Ricatti equations shows that if initial relative vorticity Q−J=∂avis sufficiently negative (anti-cyclonic), rotation does not stop wave breaking, which is taking place for any initial conditions. However, if the relative vorticity is positive (cyclonic case), as well as the derivatives of the Riemann invariants at the initial moment, there is no breaking. An example of wave breaking due to the geostrophic adjustment of the unbalanced jet is presented in Fig. 4.2.
−2L −L 0 L 2L 0
Vmax
Vjet
Fig. 4.2 Wave breaking and shock formation (right panel) during adjustment of the unbalanced jet (left panel, top to bottom: consecutive profiles of the free surface with time measured in f−1 units). Length is measured in deformation radius units: L=Rd=g Hf
4.2.1.6 “Trapped Waves” in 1.5d RSW: Pulsating Density Fronts
The above-established supra-inertiality of the spectrum of the small perturbations around a balanced 1.5d RSW front means the absence of trapped waves, and, hence, the attainability of the adjusted state by evacuating the excess of energy via inertia- gravity wave emission (eventually with shock formation). There exist, however, the RSW fronts, where the wave emission is impossible. These are the lens-type con- figurations with terminating profile of fluid height. Such RSW configurations are used to model oceanic double density fronts, either outcropping or incropping, e.g.
Griffiths et al. [10]. In Lagrangian description (4.9) the evolution of a double RSW front corresponds to positive hI terminating at x =x±. Adjustment of such fronts, therefore, should proceed without outward IGW emission. An example of adjusted front treated in literature is given in Fig. 4.3.
A family of exact unbalanced pulsating solutions is known for such fronts (Frei [9];
Rubino et al. [22]). Let us make the following ansatz:
X(x,t)=xχ(t), hI(x)= h0
2
1− x2 L2
, vI(x)=x, (4.38)
Fig. 4.3 An example of equilibrated double density front
where h0, ,L are constants. Plugging (4.38) into (4.9) and non-dimensionalizing with the timescale f−1and the length-scale L gives the following ODE forχ:
¨
χ+χ− γ
χ2 =μ, (4.39)
whereγis the Burger number fgh2L02 andμ=1+f. Integrating (4.39) once gives
˙ χ2
2 +P(χ)=E, P(χ)=χ2
2 −μχ+γ
χ, (4.40)
where the integration constant E is expressed in terms of initial conditionsχ(t = 0)
=1,χ(t˙ =0)=U :
E= U2 2 +1
2 −μ+γ. (4.41)
Equation (4.40) may be integrated in elliptic functions. The “potential” P(χ)being convex, the solution for χ is finite amplitude and oscillating with supra-inertial frequency. The minimum of P corresponds to the front in geostrophic equilibrium and constantχ=1. Thus, the adjustment (initial-value) problem for double density fronts will result, in general, in a pulsating solution, whereas relaxation to the steady state is possible only due to viscous effects (shocks).