Transitions of zonal flows in a two-layer quasi-geostrophic ocean model

Abstract

We consider a 2-layer quasi-geostrophic ocean model where the upper layer is forced by a steady Kolmogorov wind stress in a periodic channel domain, which allows to mathematically study the nonlinear development of the resulting flow. The model supports a steady parallel shear flow as a response to the wind stress. As the maximal velocity of the shear flow (equivalently the maximal amplitude of the wind forcing) exceeds a critical threshold, the zonal jet destabilizes due to baroclinic instability and we numerically demonstrate that a first transition occurs. We obtain reduced equations of the system using the formalism of dynamic transition theory and establish two scenarios which completely describe this first transition. The generic scenario is that a conjugate pair of modes loses stability and a Hopf bifurcation occurs as a result. Under an appropriate set of parameters describing related midlatitude oceanic flows, we show that this first transition is continuous: a supercritical Hopf bifurcation occurs and a stable time periodic solution bifurcates. We also investigate the case of double Hopf bifurcations which occur when four modes of the linear stability problem simultaneously destabilize the zonal jet. In this case, we prove that, in the relevant parameter regime, the flow exhibits a continuous transition accompanied by a bifurcated attractor homeomorphic to $S^3$. The topological structure of this attractor is analyzed in detail and is shown to depend on the system parameters. In particular, this attractor contains (stable or unstable) time-periodic solutions and a quasi-periodic solution.

Appendices

Appendix A: Proof of Lemma 1 and Theorem 1

We first proceed with the proof of Lemma 1 For this, we denote the adjoint modes by

$$\begin{aligned} \psi ^{*}_{m, j} = e^{i \alpha _{m} x} Y_{m}^{*}(y). \end{aligned}$$

We denote the critical eigenmode and the critical eigenvalue by

$$\begin{aligned} \psi _c = \psi _{m_c, 1}, \qquad \sigma _c = \sigma _{m_c, 1}. \end{aligned}$$

We denote the bilinear operator ${\mathcal {G}}$ as

$$\begin{aligned} {\mathcal {G}}(u) = G_{2}(u, u) \end{aligned}$$

where $G_{2}(u, v)$ is linear in each component. Let us define now

$$\begin{aligned} G_{s}(u, v) = G_{2}(u, v) + G_{2}(v, u). \end{aligned}$$

(26)

The center part of the solution is

$$\begin{aligned} u_{c} = z(t) \psi _{c} + c.c. \end{aligned}$$

(27)

where c.c. stands for complex conjugate of the terms before.

The evolution of z(t) near the onset of transition is obtained by the projection onto the critical mode $\psi _c$.

$$\begin{aligned} {\dot{z}} = \sigma _{c} z + \frac{ 1 }{ \langle {\mathcal {M}} \psi _{c}, \psi _{c}^{*} \rangle } \langle {\mathcal {G}}( u_{c} + \varPhi ), \psi _{c}^{*} \rangle . \end{aligned}$$

(28)

where $\varPhi$ is the center manifold function. We will obtain its quadratic approximation $\varPhi _2$ given by

$$\begin{aligned} \varPhi = \varPhi _2(z, {\overline{z}}) + o(2) \end{aligned}$$

Here

$$\begin{aligned} o(n) = o \left( | (z, {\overline{z}}) |^{n} \right) \end{aligned}$$

denotes higher than n-th order terms in $z, {\overline{z}}$.

Using notation (26), the reduced Eq. (28) can be written

$$\begin{aligned} {\dot{z}} = \sigma _{c} z + \frac{ 1 }{ \langle {\mathcal {M}} \psi _{c}, \psi _{c}^{*} \rangle } \langle G_{s}( u_{c}, \varPhi _2 ), \psi _{c}^{*} \rangle + o(3). \end{aligned}$$

(29)

To obtain a closed system, we need to approximate the center manifold function. The approximation of the center manifold in this case reads, see [30],

$$\begin{aligned} \begin{aligned}&\varPhi _{2} = (2 \sigma _{c} - {\mathcal {L}})^{-1} \varPi _{s} G_2(z \psi _c, z \psi _c) \\&\quad + (\sigma _c + {\overline{\sigma }}_c - {\mathcal {L}})^{-1} \varPi _{s} G_{2}\\&\quad (z \psi _{c}, \overline{z \psi _{c}}) + c.c. \end{aligned} \end{aligned}$$

(30)

where ${\mathcal {L}} = \varPi _{s} {\mathcal {M}}^{-1} {\mathcal {N}}$ and $\varPi _{s}$ is the projection on the stable space. Using formula (30), we obtain the following expansion of the center manifold (see also [5, Theorem 2])

$$\begin{aligned} \varPhi _{2} = z^{2} \sum _{j \ge 1} g_{2m_{c}, j} \psi _{2m_{c}, j} + | z |^{2} \sum _{j \ge 1} g_{0, j} \psi _{0, j} + c.c. \end{aligned}$$

(31)

Here

$$\begin{aligned} \begin{aligned}&g_{0, j} = \frac{ 1 }{ (\sigma _c + {\overline{\sigma }}_c -\sigma _{0, j}) \langle {\mathcal {M}} \psi _{0,j}, \psi _{0,j}^{*} \rangle }\\&\quad \langle G_{2}(\psi _{c}, \overline{\psi _{c}}), \psi _{0, j}^{*} \rangle \\&g_{2m_{c}, j} = \frac{ 1 }{ ( 2\sigma _{c} - \sigma _{2 m_{c}, j} ) \langle {\mathcal {M}} \psi _{2m_{c}, j}, \psi _{2m_{c},j}^{*} \rangle }\\&\quad \langle G_{2}(\psi _{c}, \psi _{c}), \psi _{2m_{c}, j}^{*} \rangle , \end{aligned} \end{aligned}$$

(32)

are the coefficients of the center manifold function.

We write (29) as (19), that is

$$\begin{aligned} {\dot{z}} = \sigma _c z + P z | z |^2 + o(3). \end{aligned}$$

which finishes the proof of Lemma 1.

Recalling the definition of $G_s$ given in (26), the transition number P can then be written as

$$\begin{aligned} P = P_0 + P_2, \end{aligned}$$

(33)

where

$$\begin{aligned} P_{0}= & {} \sum _{j \ge 1} P_{0, j}, \qquad P_{0, j} = \frac{ 1 }{ \langle {\mathcal {M}} \psi _{c}, \psi _{c}^{*} \rangle } \nonumber \\&\quad ( g_{0, j} + c.c.) \langle G_{s}(\psi _{c}, \psi _{0, j}), \psi _{c}^{*} \rangle , \end{aligned}$$

(34)

denotes the contribution of the zero-wavenumber (stable) modes $\psi _{0, j}$ while

$$\begin{aligned} P_{2}= & {} \sum _{j \ge 1} P_{2, j}, \qquad P_{2, j} = \frac{ 1 }{ \langle {\mathcal {M}} \psi _{c}, \psi _{c}^{*} \rangle } \nonumber \\&g_{2m_{c}, j} \langle G_{s}(\overline{\psi _{c}}, \psi _{2m_{c}, j}), \psi _{c}^{*} \rangle , \end{aligned}$$

(35)

denotes the contribution of the modes $\psi _{2m_{c}, j}$ on the transition number, respectively. The transition type depends on the real part of the transition number P. The proof of Theorem 1 follows from the standard Hopf bifurcation analysis of the reduced equation.

Appendix B: Proof of Lemma 2 and Theorem 2

As the reduction in the case of (18) is similar to the case of (17) given in the previous section, we will only mention the differences between these two cases. Under assumption (18), we write the center part of the solution as

$$\begin{aligned} u_c = z_1(t) \psi _1 + z_2(t) \psi _2 + c.c. \end{aligned}$$

where the first two critical modes are

$$\begin{aligned}&\psi _1 = \psi _{m_c, 1}, \quad \psi _2 = \psi _{m_c+1, 1}, \quad \psi _{-1} = \psi _{-m_c, 1}, \\&\psi _{-2} = \psi _{-m_c-1, 1}. \end{aligned}$$

with corresponding eigenvalues

$$\begin{aligned}&\sigma _1 = \sigma _{m_c, 1}, \quad \sigma _2 = \sigma _{m_c+1, 1}, \quad \sigma _{-1} = \sigma _{-m_c, 1}, \\&\sigma _{-2} = \sigma _{-m_c-1, 1} \end{aligned}$$

Eq. (29) becomes the system

$$\begin{aligned}&{\dot{z}}_j = \sigma _j z_j + \frac{ 1 }{ \langle {\mathcal {M}} \psi _j, \psi _j^* \rangle } \langle {\mathcal {G}}(u_c + \varPhi _2), \psi _j^* \rangle + o(3),\nonumber \\&j = 1,2. \end{aligned}$$

(36)

and the center manifold function (30) is replaced by

$$\begin{aligned}&\varPhi _2 = \sum _{| j |, | k |=1}^{2} z_j z_k \varPsi _{j, k} + o(2),\\&\qquad \varPsi _{j, k} = (\sigma _j + \sigma _k - {\mathcal {L}})^{-1} \varPi _s G_2(\psi _j, \psi _k), \end{aligned}$$

where $\varPi _s$ denotes the projector onto the stable subspace.

Now, Eq. (36) becomes (21) with the coefficients defined below:

$$\begin{aligned} \begin{aligned}&A = g_{1, 1, -1, 1} + g_{1, -1, 1, 1} + g_{-1, 1, 1, 1} \\&B = g_{1, 2, -2, 1} + g_{1, -2, 2, 1} + g_{2, 1, -2, 1} \\&\quad + g_{2, -2, 1, 1} + g_{-2, 1, 2, 1} + g_{-2, 2, 1, 1}\\&C = g_{2, 1, -1, 2} + g_{2, -1, 1, 2} + g_{1, 2, -1, 2} \\&\quad + g_{1, -1, 2, 2} + g_{-1, 2, 1, 2} + g_{-1, 1, 2, 2}\\&D = g_{2, 2, -2, 2} + g_{2,-2, 2, 2} + g_{-2, 2, 2 ,2}\\&g_{i, j, k, l} = \frac{ 1 }{ \langle {\mathcal {M}} \psi _l, \psi _l^* \rangle } \langle G_s( \psi _i, \varPsi _{j, k}, \psi _l^* ) \rangle , \\&\varPsi _{j ,k} = (\sigma _j + \sigma _k - {\mathcal {L}})^{-1} \varPi _s G_2(\psi _j, \psi _k) \end{aligned} \end{aligned}$$

(37)

We note that the above coefficients contain only $g_{i, j, k, l}$ for which $i+j+k = l$. The expansion of the center manifold coefficients can be written more explicitly as (see also [5, Theorem 2])

$$\begin{aligned} \varPsi _{j ,k}= & {} \sum _{i=1}^{\infty } \frac{ \langle G_2(\psi _{m_j}, \psi _{m_k}) , \psi ^*_{m_j+m_k, i} \rangle }{ \langle {\mathcal {M}} \psi _{m_j+m_k, i}, \psi ^*_{m_j+m_k, i} \rangle } \\&(\sigma _j + \sigma _k - \sigma _{m_j + m_k})^{-1} \psi _{m_j+m_k, i} \end{aligned}$$

Now we analyze Eq. (21) by first putting them in polar form

$$\begin{aligned} z_j = \rho _i e^{i \gamma _j}, \quad j = 1, 2 \end{aligned}$$

which yields

$$\begin{aligned} \begin{aligned}&\dot{ \rho _1 } = Re(\sigma _1) \rho _1 + \rho _1 (Re(A) \rho _1^2 + Re(B) \rho _2^2) + \text { h.o.t. }\\&\dot{ \rho _2 } = Re(\sigma _2) \rho _2 + \rho _2 (Re(C) \rho _1^2 + Re(D) \rho _2^2) + \text { h.o.t. } \end{aligned}\nonumber \\ \end{aligned}$$

(38)

and

$$\begin{aligned} \begin{aligned}&\dot{ \gamma _1 } = Im(\sigma _1) + \text { h.o.t. } \\&\dot{ \gamma _2 } = Im(\sigma _2) + \text { h.o.t. } \end{aligned} \end{aligned}$$

For the specific case of (22), Eq. (38) always admits the solutions which represent the periodic solutions

$$\begin{aligned}&(\rho _1, \rho _2) = \left( -\frac{ Re(\sigma _{1}) }{ Re(A) }, 0 \right) , \\&(\rho _1, \rho _2) = \left( 0, -\frac{ Re(\sigma _{2}) }{ Re(D) } \right) , \end{aligned}$$

with respective eigenvalues

$$\begin{aligned}&\kappa _1 = -2 \sigma _1, \qquad \kappa _2 = \sigma _2 - \delta \sigma _1 \\&\kappa _1 = -2 \sigma _2, \qquad \kappa _2 = \sigma _1 - \theta \sigma _2 \\ \end{aligned}$$

Also, Eq. (38) admits the following solution which represents a quasi-periodic solution

$$\begin{aligned} (\rho _1, \rho _2) = \left( \frac{ \sigma _{1} - \theta \sigma _{2} }{ Re(A) (\theta \delta - 1) }, \frac{ \sigma _{2} - \delta \sigma _{1} }{ Re(D) (\theta \delta - 1) } \right) . \end{aligned}$$

Since the Jacobian matrix of the right-hand side of (38) at the quasi-periodic solution has determinant

$$\begin{aligned} \frac{ -4 ( \sigma _2 - \delta \sigma _1 ) ( \sigma _1 - \theta \sigma _2 ) }{ \theta \delta - 1 }. \end{aligned}$$

With this information, the transition scenarios summarized in Figs. 10 and 11 can be obtained by a standard analysis. To prove the claim on the bifurcation of an $S^3$-attractor, we need to prove that $(\rho _1, \rho _2) = (0, 0)$ is locally stable equilibrium of (38) at $\varPsi = \varPsi _c$, that is when ${\text {Re}}(\sigma _1) = {\text {Re}}(\sigma _2) = 0$. In this case by assumption (22), from (38), we can obtain

$$\begin{aligned}&\frac{ \mathrm{d} }{ \mathrm{d}t } (\rho _1^2 + \rho _2^2) = Re(A) \rho _1^4 + ({\text {Re}}(B) \\&\quad + {\text {Re}}(C))\rho _1^2 \rho _2^2 + {\text {Re}}(D) \rho _2^4 < 0 \end{aligned}$$

which proves the claim.

Appendix C: Numerical treatment of the linear stability problem

To solve the eigenvalue problem numerically, we first plugin the ansatz

$$\begin{aligned}&\psi (x, y) = e^{i \alpha _{m} x} Y_{j}(y), \qquad j \in {\mathbb {N}}, m \in {\mathbb {Z}}, \quad \nonumber \\&\alpha _{m} := a m \pi . \end{aligned}$$

(39)

into the eigenvalue problem

$$\begin{aligned} \sigma {\mathcal {M}} \psi (x,y) = {\mathcal {N}}(y) \psi (x,y), \end{aligned}$$

to obtain

$$\begin{aligned} \sigma \widetilde{{\mathcal {M}}} Y(y) = \widetilde{{\mathcal {N}}}(y) Y(y), \qquad Y(y) = (Y_{1}(y), Y_{2}(y)) \end{aligned}$$

(40)

Here the linear operators $\widetilde{{\mathcal {M}}}$ and $\widetilde{{\mathcal {N}}}(y)$ are defined as

$$\begin{aligned} \widetilde{{\mathcal {M}}} = \begin{bmatrix} \varDelta _{m} - F_{1} &{} F_{1} \\ F_{2} &{} \varDelta _{m} - F_{2} \end{bmatrix}, \qquad \widetilde{{\mathcal {N}}}(y) = \begin{bmatrix} N_{11} &{} N_{12} \\ N_{21} &{} N_{22} \end{bmatrix} \end{aligned}$$

(41)

where

$$\begin{aligned} N_{11}= & {} c_{1} \cos k \pi y \left( (k \pi )^{2} + \varDelta _{m} \right) - \beta i \alpha _{m} - r_{1} \varDelta _{m} \nonumber \\ N_{12}= & {} c_{1} F_{1} \cos k \pi y \nonumber \\ N_{21}= & {} 0 \nonumber \\ N_{22}= & {} -c_{1} F_{2} \cos k \pi y - \beta i \alpha _{m} - r_{2} \varDelta _{m} \nonumber \\ \varDelta _{m}= & {} D^{2} - \alpha _{m}^{2}, \qquad D = \frac{\partial }{\partial y} \nonumber \\ c_{1}= & {} \varPsi k \pi i \alpha _{m}. \end{aligned}$$

(42)

The eigenvalue problem (40) is supplemented with the following boundary conditions

$$\begin{aligned} Y_{i}(\pm 1) = D^{2}Y_{i} (\pm 1) = 0, \qquad i=1,2. \end{aligned}$$

(43)

We use Legendre–Galerkin method to discretize and solve  (40) with boundary conditions (43). We refer to [31] for the details of the Legendre–Galerkin method and to [10] for its use in dynamical transition problems.

Let $\{ L_{j} \}$ be the Legendre polynomials and consider compact combinations of the Legendre polynomials

$$\begin{aligned} f_{j}(y) = L_{j}(y) + \sum _{k=1}^{4} c_{jk} L_{j+k}(y) \end{aligned}$$

with $c_{jk}$ chosen so that $f_{j}$ satisfy the boundary conditions (43), i.e.,

$$\begin{aligned} f_{j}(\pm 1) = D^{2}f_{j}(\pm 1) = 0. \end{aligned}$$

Table 2 Set of model parameters used in the numerical study of the problem

To discretize the eigenvalue problem, we plug

$$\begin{aligned}&Y_{i}^{N_y}(y) = \sum _{j=0}^{N_y-1} y_{j}^{(i)} f_{j}(y), \quad {\widehat{Y}}_{i} = [ y_{0}^{(i)}, \dots , y_{N_y-1}^{(i)} ]^{T}, \nonumber \\&i = 1, 2. \end{aligned}$$

(44)

into (40) to obtain

$$\begin{aligned}&\sigma \begin{bmatrix} {\widehat{\varDelta }}_{m} - F_{1} A_{3} &{} F_{1} A_{3} \\ F_{2} A_{3} &{} {\widehat{\varDelta }}_{m} - F_{2} A_{3} \end{bmatrix} \begin{bmatrix} {\widehat{Y}}_{1} \\ {\widehat{Y}}_{2} \end{bmatrix} \nonumber \\&= \begin{bmatrix} {\widehat{N}}_{11} &{} {\widehat{N}}_{12} \\ {\widehat{N}}_{21} &{} {\widehat{N}}_{22} \end{bmatrix} \begin{bmatrix} {\widehat{Y}}_{1} \\ {\widehat{Y}}_{2} \end{bmatrix} \end{aligned}$$

(45)

$$\begin{aligned}&\begin{aligned}&{\widehat{N}}_{11}= c_{1}(k \pi )^{2} A_{5} + c_{1}\left( A_{4}^{T} -\alpha _{m}^{2} A_{5}\right) \\&- \beta i \alpha _{m} A_{3} - r_{1} {\widehat{\varDelta }}_{m} \\&{\widehat{N}}_{12} = c_{1} F_{1} A_{5} \\&{\widehat{N}}_{21} = 0 \\&{\widehat{N}}_{22} = -c_{1} F_{2} A_{5}-\beta i \alpha _{m} A_{3} - r_2 {\widehat{\varDelta }}_{m} \end{aligned} \end{aligned}$$

(46)

Here

$$\begin{aligned} \begin{aligned}&A_{1} = (D^{4} f_{j}, f_{k}), \qquad A_{2} = (D^{2} f_{j}, f_{k}),\\&A_{3} = (f_{j}, f_{k}), \\&A_{4} = (\cos k\pi y D^{2}f_{j}, f_{k}), \qquad A_{5} = (\cos k \pi y f_{j}, f_{k}), \\&{\widehat{\varDelta }}_{m} = A_{2}-\alpha _{m}^{2} A_{3}, \end{aligned}\nonumber \\ \end{aligned}$$

(47)

with $(f, g) = \int _{-1}^{1} f (y) g (y) \, {\mathrm {d}}y$. The explicit expression of the matrices $A_{i},i=1,\dots ,5$ can be found in [10].

Appendix D: Practical aspects for the calculation of the transition number

The practical calculation of the $P_0$-term in (34) and the $P_2$-term in (35), boils down to the efficient calculation of the inner and trilinear products involved therein. In that respect, we provide here explicit expressions of the latter. They are given by

$$\begin{aligned} \begin{aligned}&\langle {\mathcal {M}} \psi _{m, j}, \psi _{m, j}^{*} \rangle = i \alpha _{m} \int _{-1}^{1} \left( (D^{2} - \alpha _{m}^{2} - F_{1}) Y_{m, j}^{1}\right. \\&\quad \left. + F_{1} D Y_{m, j}^{2} \right) \overline{ Y_{m, j}^{* 1}} \, {\mathrm {d}}y \\&\quad + i \alpha _{m} \int _{-1}^{1} \left( (D^{2} - \alpha _{m}^{2} - F_{2}) Y_{m, j}^{2} \right. \\&\quad \left. + F_{2} D Y_{m, j}^{1} \right) \overline{ Y_{m, j}^{* 2}} \, {\mathrm {d}}y, \end{aligned} \end{aligned}$$

and

$$\begin{aligned}&\langle G_{2} (\psi _{m,j}, \psi _{n, k}), \psi _{p, l}^{*} \rangle = -\delta _{m+n-p} \\&\left( \int _{-1}^{1} G_{2}^{1} \overline{ Y_{p,l}^{*, 1} } + G_{2}^{2} \overline{ Y_{p,l}^{*, 2} } \right) \, {\mathrm {d}}y, \\&\begin{aligned}&G_{2}^{1} = i \alpha _{m} Y_{m, j}^{1} \left( D (D^{2} - \alpha _{n}^{2} - F_{1}) Y_{n, k}^{1} + F_{1} D Y_{n,k}^{2} \right) \\&- i \alpha _{n} \left( (D^{2} - \alpha _{n}^{2} - F_{1}) Y_{n, k}^{1} + F_{1} Y_{n,k}^{2} \right) D Y_{m, j}^{1} \\&G_{2}^{2} = i \alpha _{m} Y_{m, j}^{2} \left( D (D^{2} - \alpha _{n}^{2} - F_{2}) Y_{n, k}^{2} + F_{2} D Y_{n,k}^{1} \right) \\&- i \alpha _{n} \left( (D^{2} - \alpha _{n}^{2} - F_{2}) Y_{n, k}^{2} + F_{2} Y_{n,k}^{1} \right) D Y_{m, j}^{2}. \end{aligned} \end{aligned}$$

In practice, the integrals can be evaluated by any commonly used quadrature rules in which the values of the integrand are evaluated at quadrature points. In our calculations, we use

$$\begin{aligned} \int _{-1}^{1} f(y) \, {\mathrm {d}}y = \sum _{n=0}^{N_y} f(y_{n}) \omega _{n}, \end{aligned}$$

where $y_{n}$ and $\omega _{n}$ are Legendre–Gauss–Lobatto quadrature points and weights, respectively.

Appendix E: Model parameters