Numerical study of the primitive equations in the small viscosity regime

Abstract

In this paper we study the flow dynamics governed by the primitive equations in the small viscosity regime. We consider an initial setup consisting on two dipolar structures interacting with a no slip boundary at the bottom of the domain. The generated boundary layer is analyzed in terms of the complex singularities of the horizontal pressure gradient and of the vorticity generated at the boundary. The presence of complex singularities is correlated with the appearance of secondary recirculation regions. Two viscosity regimes, with different qualitative properties, can be distinguished in the flow dynamics.

1 Intoduction

The primitive equations (PEs) are the fundamental equations governing the atmospheric and oceanic flows, see [43]. The system of the PEs is derived from the Boussinesq approximation assuming that the vertical length scale is small compared to the horizontal one. The relevant difference between the PEs and the usual Boussinesq equations lies in the fact that the momentum equation for the vertical velocity component is replaced by the hydrostatic balance for the pressure vertical gradient.

The mathematical analysis dealing with the PEs was essentially initiated in the fundamental works by Lions et al. [33,34,35], where the authors gave conditions for the existence of weak and strong solutions. These results where extended in [49] where the global existence of a weak solution was proved along with the local existence of a strong solution. In [10, 30, 31] the authors proved the existence of a unique global strong solution under different boundary conditions for the velocity components.

The analysis of the zero viscosity limit for the PEs is still challenging. In fact, when dealing with large scale motion of the ocean and considering the small scale over which viscosity acts, it is natural to neglect the viscous effect leading to the question whether the PEs converge to their inviscid counterpart, the hydrostatic Euler equation [36]. The complication arises because of the interaction with no slip boundaries and the consequent boundary layer formation. In [29] the authors investigated the vanishing viscosity problem for the PEs with no-slip boundary conditions both on the bottom and on the top, obtaining that, up to boundary layer correctors of order the square root of the viscosity, the PEs solutions converge to the hydrostatic Euler solution. The procedure required the expansion of the solution near the boundary with the inclusion of a corrector term to resolve the boundary layer and interpolate between the no slip condition and the tangential slip generated by the hydrostatic solution.

It is therefore clear that when solving the zero viscosity limit one needs also to face the mathematical and physical issues related to the hydrostatic Euler equation. In fact in [4, 25, 37] it was given evidence of a possible instability of the solution. Related to this instability is the possible blow-up of the solution. This was shown in [9], where the authors considered the 2D inviscid primitive equations and showed the finite time blow-up of the solution for a certain class of smooth initial data, and in [51] where the author considered the 3D case and gave evidence of the singularity formation by constructing self-similar solutions. An open problem is to prove whether the boundary layer corrector found in [29], develops or not singularity in finite time. In this case one could face a problem similar to the convergence of the Navier–Stokes solutions to the Prandtl solution, a long-standing problem which might be crucial to solve in order to ensure convergence of the Navier–Stokes solutions to the Euler solution outside the boundary layer. Recent results [23] have shown that the various viscous interactions developing in high Reynolds number Navier–Stokes solutions are due to the formation of complex singularities in the NS solutions.

Therefore one should expect that the convergence of the PEs solution to the hydrostatic Euler solution can be also addressed by investigating the presence of complex singularities in the PEs solution. The main aim of this paper to understand whether the PEs solutions display complex singularities, and how they move in the complex plane when viscosity becomes small. This will accomplished by performing the complex singularity tracking for two relevant quantities obtained from the PEs solution, namely the vorticity generated at the bottom and the horizontal pressure gradient. This choice is mainly dictated by the fact that these quantities are commonly used in analyzing high Reynolds number Navier–Stokes solution see [21, 23, 24, 40, 41]. We shall see that both quantities have complex singularities, and their presence is related to peculiar physical events manifesting in the flow evolution.

The rest of the paper is organized as follows. In Sect. 2 we shall introduce the primitive equations and we shall state the physical problem. In Sect. 3 we shall describe the flow evolution obtained from the numerical solution of the PEs, and we shall mainly analyze the boundary layer evolution in terms of energy and enstrophy of the flow. In Sect. 4 we shall briefly review the singularity tracking method, and we shall explain the numerical procedure used to determine the positions and the character of the singularities. In Sects. 4.1 and 4.2 we shall present the results of the singularity tracking applied to the pressure gradient and to the vorticity generated at the bottom of the domain. Finally in Sect. 5 we shall draw some conclusions.

2 Statement of the problem

The primitive equations for a velocity field $\mathbf{u}(x,y)=\left( u(x,y),v(x,y)\right)$ driven by the pressure p(x, y), where $(x,y)\in D=\mathbb {T}\times [-\,h,0]$, with $h>0$, read as

$$\begin{aligned}&\displaystyle \partial _t u+u\partial _x u+v\partial _y u+\partial _x p=\nu \left( \partial _{xx}u+\partial _{yy}u\right) , \end{aligned}$$

(1)

$$\begin{aligned}&\displaystyle \partial _xu+\partial _yv=0, \end{aligned}$$

(2)

$$\begin{aligned}&\displaystyle \partial _y p=0, \end{aligned}$$

(3)

$$\begin{aligned}&\displaystyle \gamma _D^- u=\gamma _D^- v=0, \end{aligned}$$

(4)

$$\begin{aligned}&\displaystyle \gamma _N^+ u=\gamma _D^+ v=0, \end{aligned}$$

(5)

$$\begin{aligned}&\displaystyle u(x,y,t=0)=u_0(x,y)\,, \end{aligned}$$

(6)

where we suppose that the velocity components and the pressure are periodic in $x\in \mathbb {T}$ and

$$\begin{aligned} \gamma _D^-f(x,y)= & {} f(x,y=-\,h), \end{aligned}$$

(7)

$$\begin{aligned} \gamma _D^+f(x,y)= & {} f(x,y=0), \end{aligned}$$

(8)

$$\begin{aligned} \gamma _N^+f(x,y)= & {} \partial _y f(x,y)_{|y=0}. \end{aligned}$$

(9)

In the above system, (1) is the equation for the horizontal velocity component u where $\nu$ is the kinematic viscosity of the flow, (2) is the incompressibility condition, (3) is the condition ensuring that the pressure is constant along the vertical direction, (4) and (5) are the boundary conditions for the velocity components on the bottom and on the top, and (6) is the initial condition.

To retrieve the pressure gradient in (1) we need to recover the pressure p. This is accomplished by deriving (1) with respect to the horizontal x variable and then averaging along the vertical direction y, obtaining the following equation for p:

$$\begin{aligned} \partial _{xx}p=\frac{1}{h}\left( \int _{-h}^{0}\partial _{xx}u^2d\tilde{y}-\nu [\partial _{xy}u]_{y=-h}^{y=0} \right) , \end{aligned}$$

(10)

The normal velocity is computed from (2)

$$\begin{aligned} v(x,y)=-\int _{-h}^{y}\partial _x u(x,\tilde{y}) d\tilde{y}, \end{aligned}$$

(11)

and in view of (5), we have that

$$\begin{aligned} v(x,0)=-\int _{-h}^{0}\partial _x u(x,\tilde{y}) d\tilde{y}=0. \end{aligned}$$

(12)

Condition (12) is guaranteed for all time if the horizontal and vertical components u and v of the velocity are respectively even and odd along the extended vertical domain $[-\,h,h]$. The boundary conditions (4) and (5) and the Eqs. (1) and (2) imply that u and v preserve these symmetries during the dynamics as long as it is satisfied initially. The choice of using the homogeneous Neumann BC for u at the top reflects the fact that free slip is the physical condition commonly used for atmospheric and oceanic flows. To ensure these requests, we put $\mathbb {T}=[0,2\pi ]$, and consider the following initial condition for u:

$$\begin{aligned} u_0(x,y)=\left[ \text {e}^{-\left( \frac{y-y_0}{\sigma }\right) ^2}-\text {e}^{-\left( \frac{y-(-h-y_0)}{\sigma }\right) ^2}\right] \text {e}^{-\left( \frac{x-\pi }{\sigma }\right) ^2}, \end{aligned}$$

which satisfies (4) and (5) for sufficiently small $\sigma$, and represents a pair of dipolar structures centered in $(\pi ,(-\,h-y_0))$ and $(\pi ,y_0)$, see Fig. 1a.

2.1 The numerical scheme

The whole system is solved using a spectral numerical scheme based on a fully Fourier spectral discretization for the x variable and on a collocation Chebyshev discretization for the y variable. In particular, due to the periodicity of the problem, the spatial discretization is based on following expressions for the main variables:

$$\begin{aligned} u(x,y,t)\approx & {} \sum \limits _{k=-M_x/2}^{k=M_x/2}u_{k}(y,t)e^{ikx},\quad v(x,y,t)\approx \sum \limits _{k=-M_x/2}^{k=M_x/2}v_{k}(y,t)e^{ikx}, \end{aligned}$$

(13)

$$\begin{aligned} p(x,t)\approx & {} \sum \limits _{k=-M_x/2}^{k=M_x/2}p_{k}(t)e^{ikx}, \end{aligned}$$

(14)

where $x_k=2\pi k/M_x, \quad k=0,\ldots M_x$ are the $M_x+1$ grid points in the horizontal domain. The grid points in the vertical domain are $y_j=\frac{w_jh-h}{2},\ j=0,\ldots ,M_y$, where $w_j=\cos \big (\pi (j/M_y-1)\big )$ are the collocation Gauss–Lobatto points in $[-1,1]$.

Time-integration is based on a two-stage Runge–Kutta algorithm where the discretization of the linear part is of the Crank–Nicolson type (see [44]). In particular , each mode k satisfies the following sets of equations in y:

$$\begin{aligned} u_k^{1}= & {} u^{n}_k+\Delta t\left[ N\left( u^n_k,v^n_k,p^n_k\right) +\frac{1}{2} L\left( u^n_k\right) +\frac{1}{2}L\left( u^1_k\right) \right] , \end{aligned}$$

(15)

$$\begin{aligned} u_k^{n+1}= & {} u^{n}_k+\frac{\Delta t}{2}\left[ N\left( u^n_k,v^n_k,p^n_k\right) +N\left( u^1_k,v^1_k,p^1_k\right) + L\left( u^n_k\right) + L\left( u^{n+1}_k\right) \right] , \end{aligned}$$

(16)

where the superscript indicates the $n{\text {th}}$ time step, and $\Delta t$ is the time step. In (15) and (16) the linear part is defined as

$$\begin{aligned} L(u_k)=\nu \left( -\,k^2u_k+\frac{4}{h^2}\mathbb {D}^2u_k\right) , \end{aligned}$$

where $\mathbb {D}$ is the y-derivative matrix operator. Entries of $\mathbb {D}$ are generally defined for functions approximated in the Gauss–Lobatto nodes, and are easily found in the literature of the spectral methods [3, 8, 44]. The nonlinear part is defined as

$$\begin{aligned} N(u_k,v_k,p_k)=-\left( u*u_x\right) _k-\left( \frac{2}{h}v*\mathbb {D}u\right) _k+ikp_k(t), \end{aligned}$$

where by $*$ we have denoted the (pseudo)spectral multiplication in the x variable, which is treated with the usual 3/2-method to handle aliasing effects, see [8].

Equations (15) and (16) can be recast in matrix form $\mathcal {A}U_k^{1,n+1}=F^{1,n+1}$, where $U_k^{1,n+1}=\left( u^{1,n+1}_k(y_0),\ldots ,u^{1,n+1}_k(y_{M_y})\right) ^T$, $\mathcal {A}=(1+\nu k^2)\mathcal {I}-\nu \frac{4}{h^2}\mathbb {D}^2$ with $\mathcal {I}$ the identity matrix of order $(M_y+1)\times (M_y+1)$. The vector $F^{1,n+1}$ contains in the first and last entry $\gamma _D^-u$ and $\gamma _N^+u$, and in the interior entries the terms of (15) and (16) treated explicitly. To take into account the boundary conditions (4) and (5) for the velocity component u, the first and the last row of $\mathcal {A}$ are expressed according the well known collocation formulas found for instance in [44].

In the above terms, the vertical velocity component v is retrieved from (11) through the spectral Chebyshev integration along the vertical variable (see [50]), while each mode $p_k$ is retrieved from the elliptic problem (10) according to the following straightforward spectral integration:

$$\begin{aligned} p_k(t)= & {} -\frac{\left( \int _{-h}^{0}-k^2(u*u)_kdy+ik\nu [\frac{2}{h}\mathbb {D}u_k]_{y=-h}^{y=0} \right) }{2k^2h},\quad k\ne 0 \end{aligned}$$

(17)

$$\begin{aligned} p_0(t)= & {} 0. \end{aligned}$$

(18)

We observe that the pressure is independent on the vertical variable, and therefore it is expressed only in terms of Fourier modes. Hence we do not have any spurious mode arising when we solve spectrally the elliptic problem (10) (see [2, 8] for detailed discussions on this numerical issue in the general case in which the pressure is y-dependent).

3 Physics of the flow

In this section, we shall describe the main physical events characterizing the early flow evolution governed by the PEs, where we have fixed $h=10,y_0=-2,\sigma =1/4$ in the initial condition (1314). The physical events arising during the flow evolution will be described for two different viscosity regimes, the moderate–high regime $\nu \ge 0.01$, and the low regime $\nu <0.01$. We shall mainly describe the flow in term of vorticity evolution $\omega (x,y,t)=-\,\partial _y u(x,y,t)$, vorticity generated at the bottom $\omega _b(x,t)=\omega (x,y=-\,h,t)$, pressure gradient $\partial _x p$, energy $E=\frac{1}{2}||u||_{L^2(D)}^2$ and the enstrophy $\mathcal {E}=||\omega ||_{L^2(D)}^2$. All the above quantities are relevant and have been used to describe the various interactions leading to the onset of turbulence especially in boundary layer flows (see results on high Reynolds number Navier–Stokes solutions [13, 21, 23, 27]).

Fig. 1

Vorticity $\omega$ at various time for $\nu =0.01$. The initial vorticity consists in a pair of dipolar structures centred in $y=-\,8$ and $y=-\,2$. At subsequent times vorticity is generated at the bottom, due to the no-slip boundary condition for the horizontal velocity component

Fig. 2

Vorticity $\omega$ at $t=1$ for $\nu =0.1$ and $\nu =0.01$

Fig. 3

Vorticity $\omega _b$ produced at the bottom $y=-\,h$ at various time (increments of $\Delta t=0.05$) for $\nu =0.01$, representing the moderate–high viscosity regime, and $\nu =0.001$, representing the low viscosity regime

The moderate–high viscosity regime, represented by the case $\nu =0.01$, is strongly affected by the high dissipative effects induced by the viscosity. Vorticity evolution is shown in Figs. 1 and 2b at different times. The pair of dipolar structures centered in $y=-\,8,-\,2$ are rapidly stretched along the horizontal direction. During the clockwise rotation of the dipolar structures, a boundary layer attached to the bottom forms due to the no-slip boundary condition, and two recirculation eddies having opposite rotation are generated. They are barely visible at $t=0.2$ in Fig. 1d, while the vorticity $\omega _b$ generated at the bottom is shown at different times in Fig. 3a (the eddies generated at the bottom are better visible in Fig. 2 for $\nu =0.1$ and $\nu =0.01$ at time $t=1$). One can see that $\omega _b(x)\le 0$ for $x\lesssim \pi$ while $\omega _b(x)\ge 0$ for $x\gtrsim \pi$. The different sign is justified by the sign variation in $x\approx \pi$ observed in $\partial _x p$ (see Fig. 4b at $t=0$), which accelerates flow particles for $x\lesssim \pi$ whereas decelerates them for $x\gtrsim \pi$. As time passes, the eddies thicken essentially along the horizontal direction, and spikes in $\omega _b$ and $\partial _x p$ form at $t\approx 0.1$ (see Figs. 3a, 4b). After $t\approx 0.1$, $\omega _b$ and $\partial _x p$ begin to weaken, and the eddies generated at the bottom continue to have small vertical extension and low strength. The weak vorticity production and the weakness of the eddies generated at the bottom can also be inferred by enstrophy time evolution (Fig. 5a) which, despite the vorticity generated at the bottom, is always decreasing for all time. Subsequent flow evolution shows no relevant physical phenomena, and due to the high dissipative effects the flow tends to the rest with almost vanishing energy (see Fig. 5b).

Fig. 4

a The horizontal pressure gradient $\partial _x p$ for various $\nu$ at $t=0.2$, with a magnification in the inset, b time evolution of $\partial _x p$ for $\nu =0.01$ from $t=0$ up to $t=0.25$ (time increments of $\Delta t=0.05$) and at $t=0.5$

Fig. 5

a Time evolution of the Enstrophy $\mathcal {E}$ for various $\nu$, from $t=0$ up to $t=1$. Only for the low viscosity regime ($\nu =0.002,0.001$) the vorticity production at the bottom leads to the enstrophy growth, b time evolution of the Energy E for various $\nu$, from $t=0$ up to $t=1$

The low viscosity regime, represented by the case $\nu =0.001$, shows some physical phenomena already observed in moderate–high regime. In fact a boundary layer consisting in two eddies forms at the bottom, although the lower viscosity effects allow for a higher production than in the moderate–high regime case. Secondary recirculation regions form in time, and they can be deduced by the change in the sign of $\omega _b$ observed for $t\approx 0.2$ at $x\approx 2.76,3.52$ in Fig. 3b. The secondary recirculation regions formation are mainly due to the strong variations observed in the pressure gradient $\partial _x p$. In fact in correspondence of the points where secondary recirculation regions form, the pressure gradient becomes adverse (the change of sign is visible in Fig. 4a) with respect to the flow direction between the primary recirculation region and the bottom. This variation in the pressure gradient sign is concentrated in a very narrow horizontal zone and, depending on the sign, induces a strong acceleration/deceleration which eases the formation of a new recirculation regions.

The relevant difference with respect to the moderate–high regime lies in the $\mathcal {E}$ growth observed as $\nu$ decreases (Fig. 5a), mainly due to the high vorticity production at the bottom. $\mathcal {E}$ is a main responsible of dissipation of energy (see energy evolution in Fig. 5b). In fact, standard computations allow to write the energy equation $\frac{d E(t)}{dt}=-\nu \left( \mathcal {E}(t)+||\partial _x u(t)||^2_{L^2(D)} \right)$, from which one can observe how $\mathcal {E}$ induces a decay in the energy E. After $t\ge 0.15$, when the secondary recirculation region is forming, the enstrophy begins to decrease, and no other peak is observed. The enstrophy growth has been observed in our simulations only for $\nu \le 0.002$. The enstrophy growth in the low viscosity regime is typical of the flows characterized by the formation of the boundary layer, and has been observed for a wide class of high Reynolds number boundary layer flows: for the rectilinear vortex case [21, 24], for the impulsively started disk [23], and for the dipole-vortex flow [14, 15].

4 Singularity tracking

One of the main goal of our investigation is to analyze the singular character of the solution of the PEs in the zero viscosity limit. This is accomplished by performing the singularity tracking in the complex plane. Singularity tracking has been widely used to analyze loss of analyticity in many equations arising in fluid dynamics: Euler flow [5, 6, 12, 18, 38, 42], boundary layer flow [16, 17, 20, 23, 24], Camassa-Holm equations [17], KdV equation [19, 26], vortex-sheet flow [1, 7, 22, 28, 47, 48].

Here we apply the singularity tracking to the vorticity generated at the boundary $\omega _b$ and the horizontal pressure gradient $\partial _x p$ obtained from the numerical solution of the PEs. The choice of tracking the singularities of $\omega _b, \partial _x p$ relies on the fact that in many low viscosity/high Reynolds number flows, some of the relevant physical phenomena of the boundary layer have been successfully described and understood in terms of the character of the vorticity generated on a rigid boundary and of the pressure gradient (see in particular [23] and the analysis done on the viscous-inviscid interactions ruling the zero viscosity limit in boundary layer flow).

The singularity tracking is based on the asymptotic properties of the spectrum of an analytic function. In particular, suppose that u(z) is a real function having a complex singularity in $z^{\star }=x^{\star }+i\delta$ and that $u(z)\approx (z-z^{\star })^{\alpha +i\tau }$. Using a steepest descent argument one can show that the asymptotic (in k) behaviour of the spectrum $u_k$ of u(z) is the following (see [11]):

$$\begin{aligned} u_k\approx C|k|^{-(1+\alpha )}e^{-\delta |k|}\sin (kx^{\star }+\tau \log (k)+\phi ). \end{aligned}$$

(19)

The estimation of $\delta$ and $x^*$ gives the complex location of the singularity $z^{\star }$, while determining the rate of algebraic decay $1+\alpha$ and the rate of oscillatory behaviour $\tau$ allows to classify the character of the singularity. C and $\phi$ are constants related to the strength and phase of the singularity. To apply the singularity tracking method to $\omega _b,\partial _x p$, we actually consider that they are even function, so that it is enough to apply (19) to the Fourier expansions $\omega _b(x,t)=\sum _{k\ge 0} \omega _k(t)\sin {kx}, \partial _x p(x,t)=\sum _{k\ge 0} p_k(t)\sin {kx},$ with $x\in [0,\pi ]$. Hence, for symmetry reasons, for any singularity placed $z_1^\star =x^\star +i\delta$ there will be another singularity placed in $z_2^\star =2\pi -x^\star +i\delta$ having the same character of $z_1^\star$.

To find, at each time, the parameters in (19), we consider that they should be in principle independent on the wavenumber k, although in practice they are k dependent and are better determined in specific band of k. Hence, following [1, 22], we impose that (19) holds point-wise for each k, and for each k we equate six consecutive modes $u_{k-5},u_{k-4},u_{k-3},u_{k-2},u_{k-1},u_{k}$ to the form in (19), obtaining a nonlinear system for the parameters $C,\alpha ,\delta ,x^{\star },\tau ,\phi$ whose solution returns the k-dependent values of the parameters. Specific band of k in which the parameters $C,\alpha ,\delta ,x^{\star },\tau ,\phi$ are actually k independent are typically the first 80 wavenumbers although the amplitude of this band depends also on how many modes are affected by the roundoff errors of the machine. Hence at each time we assume as values for the various parameters in (19) those obtained in the range of wavenumbers where they are k independent.

4.1 Singularity of $\partial _x p$

We first discuss the results for $\partial _x p$, and in particular how the viscosity $\nu$ influences the singularity analysis. In Fig. 6a–c we show the results coming from the numerical fitting of (19) for $k\le 80$ and for various $\nu$ at $t=0.2$, when the secondary recirculation regions have been formed for all $\nu$ considered.

We first notice that in all cases and for each time, the evaluation of the character of the complex singularities of $\partial _x p$ has always returned a real value, due to vanishing of the complex part $\tau$ of the character We observe from Fig. 6a that the character $\alpha _{\partial _x p}$ is in each case negative, compatible with a blow up of the complexification of $\partial _x p$ close $x^\star _{\partial _x p}+i\delta _{\partial _x p}$ and the consequent eruptive behaviour observed in Fig. 4a for the pressure gradient. Although $\alpha _{\partial _x p}$ is shown only for $t=0.2$, also for subsequent times its value remains essentially the same, showing some instability only when the singularity is quite far from the real domain, because in (19) the exponential decay $e^{-\delta _{\partial _x p}k}$ dominates over the algebraic decay given by $k^{-(1+\alpha _{\partial _x p})}$.

The effect of decreasing $\nu$ strongly influences the character $\alpha _{\partial _x p}$, especially when passing from the moderate–high viscosity regime to the low regime. In fact, for $\nu \le 0.002$, the character $\alpha _{\partial _x p}$ abruptly decreases with $\nu$, expressing the fact that the complex singularity has a stronger character. The real part $x^\star _{\partial _x p}$ of the complex singularity location well agrees with the position in which $\partial _x p$ has an eruptive behavior (see Fig. 4a). However the most striking effect of a decreasing viscosity shows on the distance of the singularity from the real domain: from Fig. 6c we can observe that it diminishes for decreasing $\nu$, meaning that for $\nu \rightarrow 0$ the pressure gradient becomes less regular, a condition which is compatible with the abrupt eruptive behaviour which characterizes $\partial _x p$ for small $\nu$ values.

Fig. 6

Results of the numerical fitting of (19) applied to the horizontal pressure gradient $\partial _x p$ at $t=0.2$ for various $\nu$. a Fitting of the real character $\alpha _{\partial _x p}$, b fitting of the width of analyticity $\delta _{\partial _x p}$, c fitting of the real part $x^\star _{\partial _x p}$ of the position of the singularity, d tracking in the complex plane of the singularities $x^\star _{\partial _x p}+i\delta _{\partial _x p}$ and $x^\star _{\omega _b}+i\delta _{\omega _b}$ of $\partial _x p$ and $\omega _b$ for various $\nu$ and from $t=0.05$ (right) up to $t=0.5$ (left), time increments of $\Delta t=0.05$. The two singularities share the same positions in the complex plane. For symmetry reasons there are also the singularities $2\pi -x^\star _{\partial _x p}+i\delta _{\partial _x p}$ and $2\pi -x^\star _{\omega _b}+i\delta _{\omega _b}$

In Fig. 6d we show the tracking of the complex singularity $x^\star _{\partial _x p}+i\delta _{\partial _x p}$ in the complex plane for various $\nu$ from $t=0.05$ up to $t=0.5$. At each time the singularity is closer to the real domain as $\nu$ decreases, and in all cases the singularity, after reaching a minimal distance from the real domain, begins to travel away from the real axis. This expresses the fact that, after the time in which this distance is minimal, $\partial _x p$ acquires a smoother behavior and the amplitude of its peaks diminishes as shown for instance in Fig. 4b for the case $\nu =0.01$. In all cases the time of minimal distance of the singularity from the real domain is essentially coinciding with the time when the secondary recirculation region forms in the boundary layer. This is reminiscent of what we have observed in the previous section, i.e. a smoother flow evolution after the formation of the secondary recirculation region.

4.2 Singularity of $\omega _b$

The singularity tracking method applied to $\omega _b$ has returned results that share similarities with the same analysis performed for $\partial _x p$. From the evolution of $\omega _b$ visible in Fig. 3a, b one can easily observe that the eruptive behaviour, especially visible for $\nu =0.001$, forms in the same positions where $\partial _x p$ has its main peaks (see Fig. 4a and the case $\nu =0.001$). Therefore one can expect that this reflects the presence of singularities also in $\omega _b$, having the same position of the singularities of $\partial _x p$ in complex plane. This is actually what we have obtained from the numerical evaluation of $x^\star _{\omega _b}+ix^\delta _{\omega _b}$ (and of the symmetric singularity $2\pi -x^\star _{\omega _b}+ix^\delta _{\omega _b}$): the tracking of the singularity $x^\star _{\omega _b}+ix^\delta _{\omega _b}$ in the complex plane is essentially the same of $x^\star _{\partial _x p}+ix^\delta _{\partial _x p}$, and it is shown in Fig. 6d.

Regarding the character of the singularity, also in this case the imaginary part is almost negligible, and $\alpha _{\omega _b}$ obtained from the fitting of the form (19) is shown in Fig. 7 for various $\nu$ at $t=0.2$. The obtained values are almost k-independent in the first 80 wavenumber, with values in each case negative. This reflects that the complexified $\omega _b$ blows up close to the singularity, and $\omega _b$ consequently shows the eruptive behaviour visible in Fig. 3b for $\nu =0.001$. Also for $\omega _b$ we have obtained that $\alpha _{\omega _b}$ remains almost the same for subsequent time. With respect to $\partial _x p$, $\omega _b$ has a slightly more regular behavior, due to the fact that for all $\nu$ we have obtained $\alpha _{\omega _b}\ge \alpha _{\partial _x p}$.

Fig. 7

Numerical evaluation trough (19) of the real character $\alpha _{\omega _b}$ of the main singularity of $\omega _b$ at $t=0.2$ for various $\nu$

5 Conclusions

We have computed the solutions of the two-dimensional primitive equations in moderate–low viscosity regime. Imposing the no-slip boundary condition at the bottom of the domain leads to a boundary layer development, with high vorticity production and formation of high horizontal gradient in the pressure. In the low viscosity regime, typically $\nu \le 0.002$, small scale phenomena, supporting the formation of secondary recirculation regions in the boundary layer, have been observed. These phenomena are related to the high vorticity production at the boundary, an event leading also to the global growth of the enstrophy in the early flow evolution. For moderate–high viscosity, $\nu \ge 0.01$, the evolution is characterized by a weak vorticity production, with no growth of the enstrophy in time.

The two viscosity regimes have been also analyzed in terms of complex singularities of the horizontal pressure gradient $\partial _x p$ and the vorticity $\omega _b$ generated at the bottom. The main results is that both quantities have complex singularities which are closer to the real domain as $\nu$ decreases. All the singularities tracked show negative characters, stronger in the low viscosity regime. This reflects the eruptive behavior observed in $\partial _x p$ and $\omega _b$. In al cases the singularities reach, at a specific time, a minimal distance from the real domain and, after this event, they begin to depart from the real domain. This expresses that the width of the analyticity of the solution becomes larger, and the overall flow evolution becomes smoother; it is easy to guess that, asymptotically, the flow reaches a rest state, an event which can be predicted by the decay of the energy.

The numerical solutions we have obtained share some similarities with the high Reynolds number Navier–Stokes solutions analyzed in different configurations, see [15, 21, 23, 39,40,41]. In fact, high Reynolds number flows present the typical small scale phenomena here observed in the PEs solutions, as well as the presence of complex singularities in the solution which are precursor of the interactions between the boundary layer and inviscid outer flow. In [23] it was reported a comparison between the complex singularities of the Navier–Stokes solution and the Prandtl singularity, showing how NS solutions have a much richer singular structure which, apparently, cannot be approximated by the Prandtl singularity. We plan to perform a similar investigation for the PEs. Moreover, it would be interesting to study the asymptotic behavior of the solutions of the PEs using the Fourier Splitting technique, see [45], in analogy to what recently proposed in [46] for the shallow water model with varying bottom topography proposed in [32]. This is a work in progress that will appear elsewhere.